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NZ Scholarship Statistics study materials
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NZ Scholarship Statistics practice questions

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Question 1Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A random variable X has E(X) = 12 and Var(X) = 9. A new variable Y is defined by Y = 5 - 2X. What is Var(Y)?

  1. 36 Correct answer.
  2. 9
  3. 18
  4. 4

Explanation

For a linear transformation Y = a + bX, the variance is Var(Y) = b squared times Var(X). Adding a constant shifts the distribution but does not change spread, so the 5 is irrelevant. Here b = -2, so Var(Y) = 4 x 9 = 36.

Question 2Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A hypothesis test returns a p-value of 0.03 at the 5% significance level. Which interpretation of this p-value is correct?

  1. If the null hypothesis were true, a test statistic at least as extreme as the one observed would occur about 3% of the time. Correct answer.
  2. There is a 97% chance the alternative hypothesis is true.
  3. The probability of a Type I error in this specific test is 0.03.
  4. There is a 3% chance the null hypothesis is true.

Explanation

A p-value is the probability of obtaining a result at least as extreme as that observed, calculated under the assumption that the null hypothesis holds. It is a conditional probability of the data given the null, not of the null given the data. The significance level, not the p-value, fixes the Type I error rate for the procedure.

Question 3Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A sample of size n = 100 has sample proportion p-hat = 0.40. Using the normal approximation, what is the standard error of p-hat?

  1. 0.0024
  2. 0.60
  3. 0.24
  4. 0.049 Correct answer.

Explanation

The standard error of a sample proportion is the square root of p-hat(1 - p-hat)/n. Substituting gives the square root of (0.40 x 0.60)/100 = 0.0024. The square root of 0.0024 is approximately 0.049.

Question 4Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a designed experiment, what is the main purpose of random allocation of treatments to units?

  1. To balance the effects of unmeasured variables across treatment groups. Correct answer.
  2. To increase the sample size available for analysis.
  3. To remove the need for a control group.
  4. To ensure the sample is representative of the wider population.

Explanation

Random allocation spreads the influence of unknown or unmeasured confounding variables roughly evenly across treatment groups, so differences can be attributed to the treatment. It addresses internal validity rather than sample size or population representativeness. A control group is still needed to provide a comparison baseline.

Question 5Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Which statement best describes the relationship between a confidence interval and a two-sided hypothesis test at the same significance level?

  1. A 95% confidence interval contains all plausible values of the parameter that would not be rejected at the 5% level. Correct answer.
  2. A 95% confidence interval contains all values that would be rejected at the 5% level.
  3. The width of the confidence interval equals the p-value of the test.
  4. A confidence interval and a hypothesis test always give contradictory conclusions.

Explanation

A (1 - alpha) confidence interval consists of exactly those parameter values that a two-sided test at level alpha would fail to reject. This duality means the interval and the test always agree on whether a hypothesised value is plausible. The interval width depends on standard error and the critical value, not the p-value.

Question 6Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A data set has a strongly right-skewed distribution. Which measure pair is most appropriate for summarising its centre and spread?

  1. Mode and variance
  2. Median and interquartile range Correct answer.
  3. Mean and standard deviation
  4. Mean and interquartile range

Explanation

For skewed data the median and interquartile range are resistant to the influence of extreme values in the long tail. The mean and standard deviation are pulled toward the tail and can misrepresent the typical value. Pairing a resistant centre with a resistant spread is the consistent choice.

Question 7Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a scatter plot of two quantitative variables, the correlation coefficient is r = -0.85. Which conclusion is justified?

  1. The relationship between the variables is strongly non-linear.
  2. About 85% of the variation in one variable is explained by a linear relationship with the other.
  3. Increases in one variable cause decreases in the other.
  4. There is a strong negative linear association between the two variables. Correct answer.

Explanation

A correlation of -0.85 indicates a strong negative linear association, meaning that as one variable increases the other tends to decrease. Correlation does not establish causation. In simple least-squares regression with an intercept, r squared is 0.7225, so about 72% of sample response variation is accounted for by the linear fit.

Question 8Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls equals 7?

  1. 1/36
  2. 1/6 Correct answer.
  3. 7/36
  4. 1/12

Explanation

There are 36 equally likely ordered outcomes. The sums equal to 7 arise from (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), which is 6 outcomes. Thus the probability is 6/36 = 1/6.

Question 9Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A sample of size n = 16 is drawn from a normal population with unknown mean and unknown variance. Which distribution should be used for inference about the population mean?

  1. Standard normal distribution
  2. F distribution with 15 and 15 degrees of freedom
  3. Chi-squared distribution
  4. t distribution with 15 degrees of freedom Correct answer.

Explanation

When the population standard deviation is estimated from a small sample and the population is normal, the standardised sample mean follows a t distribution. The degrees of freedom equal n - 1, which is 15 here. The normal distribution would require a known population standard deviation or a large sample.

Question 10Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A researcher tests whether a new tutoring programme raises mean test scores. The null hypothesis is that the mean is unchanged. A Type II error occurs when:

  1. the significance level is set too low
  2. the sample mean differs from the population mean
  3. the null hypothesis is rejected when it is actually true
  4. the null hypothesis is not rejected when it is actually false Correct answer.

Explanation

A Type II error is failing to reject a false null hypothesis, meaning a real effect is missed. Its probability is often denoted beta, and power is 1 - beta. A Type I error is the reverse: rejecting a true null hypothesis.

Question 11Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Two independent samples give sample means 50 and 46 with standard errors of the difference equal to 2. What is the approximate 95% confidence interval for the difference in population means, using a critical value of 1.96?

  1. (0.08, 7.92) Correct answer.
  2. (2.08, 5.92)
  3. (3.0, 5.0)
  4. (1.04, 6.96)

Explanation

The observed difference is 50 - 46 = 4. The margin of error is 1.96 x 2 = 3.92. Subtracting and adding gives the interval from 0.08 to 7.92. Because the interval excludes zero, the difference would be significant at the 5% level.

Question 12Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a simple linear regression of y on x, the residual plot shows a clear U-shaped curve. What does this most likely indicate?

  1. The relationship between x and y is non-linear and a linear model may be inadequate. Correct answer.
  2. The residuals are normally distributed.
  3. The assumption of constant variance is satisfied.
  4. The linear model is appropriate because residuals are small.

Explanation

A systematic curved pattern in the residuals indicates that the linear term does not capture the relationship, so a non-linear term may be needed. Random scatter with no pattern supports the linear model. A U-shape is a signal of model inadequacy rather than of normality or constant variance.

Question 13Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A survey of 400 voters finds 220 support a candidate. What is the approximate 95% confidence interval for the population proportion, using a critical value of 1.96?

  1. (0.52, 0.58)
  2. (0.48, 0.62)
  3. (0.50, 0.60) Correct answer.
  4. (0.53, 0.57)

Explanation

The sample proportion is 220/400 = 0.55. The standard error is the square root of (0.55 x 0.45)/400, approximately 0.0249. The margin of error is 1.96 x 0.0249, about 0.049, giving the interval from about 0.50 to 0.60.

Question 14Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A boxplot shows a distribution with a long upper whisker and many high outliers. Which combination of features is consistent with this display?

  1. Mean equal to median and symmetry
  2. Mean greater than median and positive skew Correct answer.
  3. Mean less than median and negative skew
  4. Median greater than upper quartile

Explanation

A long upper whisker with high outliers indicates a right-skewed distribution. In right-skewed data the mean is pulled toward the tail and typically exceeds the median. A negative skew would show a long lower whisker instead.

Question 15Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A test has a power of 0.80 for detecting a specified effect. What does this mean?

  1. There is an 80% chance of rejecting the null hypothesis when the specified effect is present. Correct answer.
  2. There is an 80% chance of a Type I error.
  3. There is an 80% chance the null hypothesis is true.
  4. There is an 80% chance the sample is representative.

Explanation

Power is the probability of correctly rejecting a false null hypothesis, so a power of 0.80 means an 80% chance of detecting the specified effect when it truly exists. It complements the Type II error rate, which would be 0.20 here. Power does not describe the truth of the null or the representativeness of the sample.

Question 16Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a chi-squared test of independence for a 3 x 4 contingency table, how many degrees of freedom are used?

  1. 5
  2. 12
  3. 6 Correct answer.
  4. 11

Explanation

The degrees of freedom for a test of independence are (r - 1)(c - 1), where r and c are the numbers of rows and columns. Here (3 - 1)(4 - 1) = 2 x 3 = 6. The total number of cells, 12, is not the degrees of freedom.

Question 17Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Which statement about the expected value of a discrete random variable is correct?

  1. It is the sum of each value multiplied by its probability. Correct answer.
  2. It is always equal to the median.
  3. It must be one of the possible values of the variable.
  4. It measures the spread of the distribution.

Explanation

The expected value is the probability-weighted average of all possible values, computed as the sum of value times probability. It need not be an attainable value; a die roll has expected value 3.5, which is not a face. Spread is measured by variance, and the expected value need not equal the median for skewed distributions.

Question 18Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A study finds a statistically significant difference at the 1% level, but the estimated effect is very small in practical terms. What is the most reasonable interpretation?

  1. The result must be due to a Type I error.
  2. Statistical significance guarantees practical importance.
  3. The sample size was too small to detect the effect.
  4. The effect is real but may have limited practical importance, and effect size should be reported. Correct answer.

Explanation

Statistical significance indicates that an effect is unlikely under the null hypothesis, not that it is large. With large samples, small effects can be detected reliably. Reporting effect sizes and confidence intervals helps judge practical relevance.

Question 19Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?

  1. 0.5
  2. 0.9
  3. 0.2
  4. 0.7 Correct answer.

Explanation

The addition rule states P(A or B) = P(A) + P(B) - P(A and B). Substituting gives 0.5 + 0.4 - 0.2 = 0.7. Because P(A and B) equals P(A)P(B), the events are independent, but the union rule is unchanged.

Question 20Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A sampling distribution of the sample mean is described as having a standard error that decreases as sample size increases. Which relationship is correct?

  1. Standard error is inversely proportional to the square root of n. Correct answer.
  2. Standard error is independent of n.
  3. Standard error is proportional to the square root of n.
  4. Standard error is proportional to n.

Explanation

The standard error of the mean equals the population standard deviation divided by the square root of n. Thus it is inversely proportional to the square root of n, so quadrupling the sample size halves the standard error. It is not proportional to n or to its square root directly.

Question 21Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A randomised block design is used instead of a completely randomised design. What is the primary benefit?

  1. It reduces variability due to the blocking factor, improving precision of treatment comparisons. Correct answer.
  2. It guarantees the sample is representative of the population.
  3. It removes the need for replication within blocks.
  4. It allows causal conclusions without randomisation.

Explanation

Blocking groups similar units together so that variation attributable to the blocking factor is removed from the comparison of treatments, increasing precision. Randomisation of treatments within blocks is still required for valid causal inference. Replication within blocks remains important.

Question 22Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A regression model has R squared = 0.92, but the residual standard deviation is large relative to the response values. Which explanation is most plausible?

  1. The residuals must be non-normal.
  2. The model must have a negative slope.
  3. R squared is irrelevant when the residual spread is large.
  4. The response variable has a large total variance, so a high R squared can coexist with substantial residual spread. Correct answer.

Explanation

R squared is the proportion of total variation explained, so if the response has a large total variance, a high proportion explained can still leave a sizeable absolute residual spread. R squared is a relative measure and should be read alongside the residual standard deviation. Neither the slope sign nor residual normality follows from this pattern.

Question 23Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A sequence of independent Bernoulli trials each has success probability p. Let X be the number of trials until the first success. Which statement about X is correct?

  1. X has a binomial distribution with parameters n and p.
  2. X has a Poisson distribution with mean p.
  3. X has a normal distribution for large p.
  4. X has a geometric distribution with mean 1/p. Correct answer.

Explanation

The number of trials until the first success in independent Bernoulli trials follows a geometric distribution. Its mean is 1/p, reflecting the expected waiting time. The binomial distribution counts successes in a fixed number of trials, and the Poisson counts events in an interval.

Question 24Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A 95% confidence interval for a mean is (10.2, 15.8). A statistician claims that a 99% confidence interval computed from the same data will also contain 12. Which statement is correct?

  1. It will be narrower than the 95% interval and may exclude 12.
  2. It will be wider than the 95% interval and, since the intervals share the same centre, it must contain 12. Correct answer.
  3. The 99% interval cannot be compared with the 95% interval.
  4. It is guaranteed to contain 12 because 12 lies inside the 95% interval.

Explanation

A higher confidence level requires a larger critical value, so the 99% interval is wider than the 95% interval and shares the same sample centre. Since 12 lies inside the narrower interval centred at 13, it also lies inside the wider interval with the same centre. This reflects the nesting of confidence intervals at increasing levels.

Question 25Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a two-sample t-test comparing means, the assumption of equal variances is questionable. Which approach is most appropriate?

  1. Abandon inference entirely.
  2. Use a paired t-test even though the samples are independent.
  3. Use a Welch t-test, which does not assume equal variances. Correct answer.
  4. Proceed with the pooled-variance t-test regardless of the evidence.

Explanation

The Welch t-test adjusts the degrees of freedom and does not assume equal population variances, making it robust when variances differ. The pooled-variance test is only appropriate when equal variances are reasonable. A paired test requires matched or repeated observations, which is not the case here.

Question 26Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A statistician reports a 95% confidence interval for a population proportion as 0.42 ± 0.03. Which interpretation of this interval is correct?

  1. In repeated sampling, about 95% of intervals constructed this way would contain the true proportion. Correct answer.
  2. The interval contains 95% of the population values.
  3. About 95% of sample proportions fall between 0.39 and 0.45.
  4. There is a 95% probability that the true proportion lies between 0.39 and 0.45.

Explanation

The confidence level describes the long-run capture rate of the procedure: if the same sampling and interval construction were repeated many times, roughly 95% of the resulting intervals would contain the fixed but unknown parameter. Once a specific interval is computed, the parameter is either inside it or not, so probability statements about that single realised interval are not strictly correct.

Question 27Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A random variable X has the discrete uniform distribution on the values 1, 2, 3, 4, 5, 6. What is Var(X)?

  1. 12.25
  2. 3.5
  3. 6
  4. 35/12 Correct answer.

Explanation

For a discrete uniform distribution on 1 to n, the variance is (n squared minus 1) divided by 12. With n = 6 this gives 35/12, about 2.92. The mean is 3.5, but variance measures average squared deviation, not the mean itself.

Question 28Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a hypothesis test, the p-value is 0.03 and the significance level is 0.05. Which conclusion is justified?

  1. The effect size is large in practical terms.
  2. The null hypothesis is false with probability 0.03.
  3. The probability that the result is due to chance is 0.05.
  4. There is sufficient evidence to reject the null hypothesis at the 5% level. Correct answer.

Explanation

A p-value of 0.03 is below the 0.05 threshold, so the result falls in the rejection region and the null hypothesis is rejected. The p-value is the probability of data at least as extreme as observed, assuming the null is true, not the probability that the null is false.

Question 29Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A scatter plot shows a strong curved relationship between two variables, with points forming a clear arch. The Pearson correlation coefficient is computed and found to be close to zero. What is the best explanation?

  1. The correlation coefficient only measures the strength of linear association, so a strong nonlinear pattern can still give r near zero. Correct answer.
  2. The data must contain an outlier that distorts the result.
  3. The calculation must be wrong because the pattern is strong.
  4. The variables are unrelated because correlation is near zero.

Explanation

Pearson correlation quantifies the strength and direction of a linear relationship only. A pronounced but non-linear pattern such as an arch can produce an r close to zero even though the variables are strongly dependent, so r near zero does not imply independence.

Question 30Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Which statement about the standard error of the sample mean is correct?

  1. It equals the population standard deviation divided by the square root of the sample size. Correct answer.
  2. It equals the sample mean divided by the sample size.
  3. It equals the population standard deviation regardless of sample size.
  4. It increases as the sample size increases.

Explanation

The standard error of the sample mean is sigma divided by the square root of n. This shows that the sampling variability of the mean shrinks as the sample size grows, at a rate proportional to the inverse square root of n.

Question 31Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A data set of house prices contains one property priced far above all the others. Which pair of summary statistics is most resistant to this single extreme value?

  1. Mean and range
  2. Median and interquartile range Correct answer.
  3. Standard deviation and range
  4. Mean and standard deviation

Explanation

The median and interquartile range are based on positional summaries and ignore the magnitude of extreme values, so they stay stable when one observation is very large. Mean, standard deviation and range all use the extreme value directly and are pulled or inflated by it.

Question 32Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Two events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.5. What is P(A and B)?

  1. 0.2
  2. 0 Correct answer.
  3. 0.8
  4. 0.15

Explanation

Mutually exclusive events cannot occur together, so their intersection has probability zero. The addition rule then gives P(A or B) = 0.3 + 0.5 = 0.8, but the joint probability asked for here is zero.

Question 33Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A sampling frame lists only households with a landline telephone, and a survey selects from this list. Which type of bias is most directly introduced?

  1. Measurement bias
  2. Undercoverage bias Correct answer.
  3. Non-response bias
  4. Recall bias

Explanation

Undercoverage occurs when some members of the target population have no chance of being selected because the sampling frame omits them. Households without a landline are excluded entirely, so the frame does not represent the whole population.

Question 34Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A quality inspector records the number of defective items in samples of 50 from a production line. The defect rate is small. Which probability model is most appropriate for the count of defectives in a sample?

  1. Binomial distribution Correct answer.
  2. Exponential distribution
  3. Normal distribution
  4. Poisson distribution

Explanation

The count of defectives in a fixed number of independent trials, each with the same defect probability, is binomial. A Poisson model can approximate this when n is large and p is small, but the exact structure of a fixed sample of 50 with constant probability is binomial.

Question 35Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A researcher wants to test whether three different fertilisers produce different mean yields, using independent samples from each. Which test is most appropriate?

  1. A paired t-test
  2. A one-way analysis of variance Correct answer.
  3. A two-sample t-test applied to the two largest groups
  4. A chi-squared test of independence

Explanation

Comparing means of three or more independent groups is the setting for one-way analysis of variance, which tests the null hypothesis of equal population means. Repeated pairwise t-tests would inflate the Type I error rate.

Question 36Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a simple linear regression the slope estimate is 2.4 with a standard error of 0.8. What is the value of the t statistic for testing whether the slope is zero?

  1. 0.33
  2. 1.92
  3. 3.0 Correct answer.
  4. 1.6

Explanation

The t statistic for a slope is the estimate divided by its standard error, so 2.4 divided by 0.8 equals 3.0. This measures how many standard errors the estimate lies from the hypothesised value of zero.

Question 37Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A random sample of 25 observations from a normal population yields a sample mean of 30 and a sample standard deviation of 5. What is the 95% confidence interval for the population mean?

  1. 30 ± 1.96 × 1.0
  2. 30 ± 2.064 × 5.0
  3. 30 ± 2.064 × 1.0 Correct answer.
  4. 30 ± 1.96 × 5.0

Explanation

The population standard deviation is unknown and the sample is small, so the t distribution with 24 degrees of freedom is used; the critical value is about 2.064. The standard error is 5 divided by the square root of 25, which equals 1.0.

Question 38Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A test has power 0.85 for a specified alternative. If the sample size is increased while the significance level stays fixed, what happens to the power for the same alternative?

  1. It becomes equal to the significance level.
  2. It increases, because the standard error of the test statistic shrinks. Correct answer.
  3. It stays exactly the same, because power depends only on the significance level.
  4. It decreases, because larger samples are more variable.

Explanation

Increasing the sample size reduces the standard error of the estimator, so the distribution under the alternative separates further from the null distribution. For a fixed significance level this raises the probability of rejecting a false null, that is, the power.

Question 39Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A die is rolled repeatedly. Let X be the number of rolls until the first six appears. Which distribution does X follow?

  1. Poisson with mean one sixth
  2. Geometric with success probability one sixth Correct answer.
  3. Uniform on the values 1 to 6
  4. Binomial with n trials and p equal to one sixth

Explanation

The geometric distribution models the number of independent Bernoulli trials needed to obtain the first success, with probability mass (5/6) to the power k minus 1 times 1/6. The binomial instead counts successes in a fixed number of trials.

Question 40Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

In a multiple regression with four predictor variables and 40 observations, how many degrees of freedom are associated with the residual sum of squares?

  1. 35 Correct answer.
  2. 39
  3. 4
  4. 40

Explanation

The residual degrees of freedom equal the number of observations minus the number of estimated parameters, including the intercept. With 40 observations and five parameters (intercept plus four slopes), the residual degrees of freedom are 35.

Question 41Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A study reports an odds ratio of 2.5 with a 95% confidence interval of 1.1 to 5.7 for a disease associated with an exposure. Which interpretation is correct?

  1. The association is not statistically significant because the interval is wide.
  2. The risk of disease is exactly 2.5 times higher for the exposed group.
  3. The exposure is associated with significantly higher odds of disease at the 5% level, since the interval excludes 1. Correct answer.
  4. The exposure causes the disease with probability 2.5.

Explanation

For an odds ratio, the null value is 1, and an interval that excludes 1 indicates a statistically significant association at the 5% level. The interval is wide, reflecting imprecision, but it still lies entirely above 1.

Question 42Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A survey uses stratified sampling with strata defined by age group. Why is stratification often preferred to simple random sampling here?

  1. It guarantees that the sample mean equals the population mean.
  2. It eliminates all non-sampling error.
  3. It can reduce the variance of estimates by ensuring representation of each age group. Correct answer.
  4. It removes the need for random selection within strata.

Explanation

Stratification divides the population into homogeneous groups and samples within each, which typically reduces the variability of the overall estimate compared with simple random sampling. It still requires random selection within strata and does not by itself remove non-sampling errors.

Question 43Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A test statistic is computed as z = 1.8 for a two-sided test. Using the standard normal distribution, what is the approximate p-value?

  1. 0.036
  2. 0.072 Correct answer.
  3. 0.180
  4. 0.964

Explanation

For a two-sided test the p-value is twice the upper-tail probability beyond the absolute value of the statistic. The upper tail beyond 1.8 is about 0.036, so doubling gives approximately 0.072.

Question 44Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A residual plot from a fitted regression shows residuals fanning out as the fitted values increase. Which assumption is most clearly violated?

  1. Constant variance of errors Correct answer.
  2. Normality of errors
  3. Independence of errors
  4. Linearity of the relationship

Explanation

A fan shape indicates that the spread of residuals changes with the fitted values, which violates the constant-variance (homoscedasticity) assumption. Linearity would be questioned by a systematic curve, and normality by the shape of the residual histogram.

Question 45Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A random variable X takes value 0 with probability 0.7 and value 10 with probability 0.3. What is the standard deviation of X?

  1. 21
  2. 3.0
  3. 4.58 Correct answer.
  4. 2.1

Explanation

The mean is 0 times 0.7 plus 10 times 0.3, which equals 3. The variance is 0.7 times (0 minus 3) squared plus 0.3 times (10 minus 3) squared, which equals 6.3 plus 14.7, or 21, so the standard deviation is the square root of 21, about 4.58.

Question 46Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A clinical trial compares a new treatment with a placebo, and the primary outcome is the proportion of patients who recover. Which test is most appropriate for comparing the two proportions?

  1. A paired t-test on before and after measurements
  2. A one-sample z-test for a single proportion
  3. A chi-squared test for association in a two-by-two table Correct answer.
  4. A two-sample t-test on the recovery proportions

Explanation

Two independent groups with a binary outcome give a two-by-two table of counts, and a chi-squared test of association, equivalent to a two-proportion z-test, assesses whether the recovery proportions differ. A t-test is designed for continuous means rather than proportions.

Question 47Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

Let X have a normal distribution with mean 50 and standard deviation 10. What is the probability that X exceeds 65?

  1. About 0.9332
  2. About 0.0668 Correct answer.
  3. About 0.4332
  4. About 0.1587

Explanation

Standardising gives z equal to (65 minus 50) divided by 10, which is 1.5. The upper-tail probability beyond 1.5 is about 0.0668, so roughly 6.7% of values exceed 65.

Question 48Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

An experiment compares four treatments and uses a one-way analysis of variance. The F statistic is large and the null hypothesis is rejected. What is the appropriate next step?

  1. Conclude that the largest sample mean is significantly the largest.
  2. Conclude that all four treatment means differ from each other.
  3. Increase the sample size until the F statistic is even larger.
  4. Conduct post-hoc pairwise comparisons with a multiple-comparison adjustment. Correct answer.

Explanation

A significant F test indicates that not all population means are equal, but it does not identify which pairs differ. Post-hoc procedures with an adjustment for multiple comparisons are used to explore specific pairwise differences while controlling the error rate.

Question 49Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A regression of y on x gives a coefficient of determination of 0.81. Which statement is correct?

  1. 81% of the variation in y is explained by its linear regression on x. Correct answer.
  2. 81% of the observations lie exactly on the fitted line.
  3. The correlation between x and y is 0.81.
  4. The slope of the regression line equals 0.81.

Explanation

The coefficient of determination measures the proportion of total variation in the response explained by the fitted linear model, so 0.81 means 81% of the variation in y is accounted for by regression on x. The correlation magnitude would be the square root of 0.81, namely 0.9.

Question 50Statistics (NZ Scholarship Statistics subject, content domains not enumerated in source)

A test of a null hypothesis is conducted at the 5% significance level. Unknown to the researcher, the null hypothesis is actually true. Which outcome is possible?

  1. A Type II error, failing to reject a false null hypothesis
  2. A correct decision, always failing to reject the null hypothesis
  3. A power failure, since power is zero when the null is true
  4. A Type I error, rejecting a true null hypothesis Correct answer.

Explanation

When the null hypothesis is true, rejecting it is a Type I error, and its probability is controlled at the significance level. A Type II error can only occur when the null hypothesis is false, so it is not possible in this scenario.