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NCEA Level 3 Statistics study materials
NCEA Level 3 Statistics study resources

NCEA Level 3 Statistics practice questions

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Question 1Statistics

A random variable X counts the number of sixes obtained when a fair die is rolled 10 times. Which distribution does X follow, and what are its parameters?

  1. Binomial with n = 10 and p = 1/6 Correct answer.
  2. Binomial with n = 6 and p = 1/10
  3. Poisson with λ = 10/6
  4. Normal with mean 10/6 and variance 50/36

Explanation

X counts the number of successes in a fixed number of 10 independent trials, each with constant success probability 1/6, which is exactly the binomial setup. So X ~ Binomial(n = 10, p = 1/6).

Question 2Statistics

A sample of 64 observations has sample mean x̄ = 42 and sample standard deviation s = 8. A 95% confidence interval for the population mean is calculated. Which value is the standard error used in this interval?

  1. 1 Correct answer.
  2. 8
  3. 0.125
  4. 2

Explanation

The standard error of the sample mean is s/√n = 8/√64 = 8/8 = 1. This quantity measures the variability of x̄ across repeated samples and is the multiplier base for the confidence interval.

Question 3Statistics

In a hypothesis test the p-value is 0.032. Which interpretation of this p-value is correct?

  1. If the null hypothesis were true, the probability of obtaining a result at least as extreme as that observed is 0.032. Correct answer.
  2. 3.2% of the sample data are inconsistent with the null hypothesis.
  3. There is a 3.2% probability that the alternative hypothesis is false.
  4. There is a 3.2% probability that the null hypothesis is true.

Explanation

A p-value is computed assuming the null hypothesis is true; it is the probability of a test statistic at least as extreme as the observed one. It is not the probability that a hypothesis is true or false.

Question 4Statistics

A scatterplot of variables x and y shows a strong linear relationship. The least-squares regression line is ŷ = 12.5 + 3.2x. What is the predicted value of y when x = 10?

  1. 44.5 Correct answer.
  2. 32.0
  3. 15.7
  4. 125.0

Explanation

Substituting x = 10 into the equation gives ŷ = 12.5 + 3.2 × 10 = 12.5 + 32 = 44.5. The intercept 12.5 is the predicted y when x equals zero.

Question 5Statistics

A researcher wants to estimate the mean time students spend on homework each week. Which sampling method is most likely to produce an unbiased estimate?

  1. Surveying the first 50 students who arrive at school on Monday.
  2. Selecting students using a random number generator from a list of all enrolled students. Correct answer.
  3. Surveying students who volunteer after an announcement.
  4. Surveying students in the library during study period.

Explanation

A simple random sample drawn from a complete list gives every student an equal chance of selection, which removes selection bias. The other methods over-represent particular groups, such as volunteers or early arrivals.

Question 6Statistics

A dataset of 200 values has a mean of 50 and a median of 62. Which description best fits the shape of the distribution?

  1. Negatively skewed (skewed left) Correct answer.
  2. Symmetric
  3. Bimodal
  4. Positively skewed (skewed right)

Explanation

When the mean is less than the median, the long tail of the distribution lies to the left, pulling the mean downward. This indicates negative skew.

Question 7Statistics

Events A and B satisfy P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2. What is P(A ∪ B)?

  1. 0.1
  2. 0.2
  3. 0.7 Correct answer.
  4. 0.9

Explanation

Using the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.7. The intersection is subtracted because it is counted twice.

Question 8Statistics

A 95% confidence interval for a population proportion is (0.42, 0.58). Which statement is a correct interpretation?

  1. 95% of repeated samples will give a sample proportion between 0.42 and 0.58.
  2. 95% of the population proportions lie between 0.42 and 0.58.
  3. There is a 95% probability that the sample proportion lies in this interval.
  4. We are 95% confident that the population proportion lies between 0.42 and 0.58. Correct answer.

Explanation

Confidence refers to the long-run performance of the method: 95% of intervals constructed this way would capture the true parameter. For the single interval obtained, we state 95% confidence that it contains the population proportion.

Question 9Statistics

The heights of adult males are approximately normally distributed with mean 175 cm and standard deviation 7 cm. What proportion of males are taller than 189 cm?

  1. About 0.16
  2. About 0.025 Correct answer.
  3. About 0.32
  4. About 0.05

Explanation

189 cm is two standard deviations above the mean, since (189 − 175)/7 = 2. For a normal distribution about 95% of values lie within two standard deviations, so roughly 2.5% lie above +2 SD.

Question 10Statistics

A study finds a correlation of r = −0.78 between hours of television watched and scores on a fitness test. Which conclusion is justified?

  1. There is a strong negative linear association between television hours and fitness scores. Correct answer.
  2. About 78% of the variation in fitness scores is explained by television hours.
  3. Reducing television viewing will improve fitness scores.
  4. Watching television causes lower fitness scores.

Explanation

A correlation of −0.78 indicates a strong negative linear association between the two variables. Correlation alone cannot establish causation or the direction of any causal effect.

Question 11Statistics

A quality controller inspects 20 items from a large batch. Historically 5% of items are defective. Which distribution models the number of defective items in the sample?

  1. Normal with mean 1 and standard deviation 1
  2. Uniform with 20 outcomes
  3. Binomial with n = 20 and p = 0.05 Correct answer.
  4. Poisson with λ = 20

Explanation

There are a fixed 20 independent inspections with a constant defect probability of 0.05, so the count is binomial. The expected number of defects is np = 1, but the distribution is binomial, not Poisson or normal.

Question 12Statistics

In a two-way table of 400 observations, a chi-square test for independence gives a test statistic of 9.49 with 4 degrees of freedom. At the 5% significance level, what is the decision?

  1. Fail to reject the null hypothesis because 9.49 is less than the critical value of 9.49. Correct answer.
  2. Reject the null hypothesis because 9.49 exceeds the critical value of 3.84.
  3. Reject the null hypothesis only if the sample size exceeded 1000.
  4. Accept the null hypothesis as true with certainty.

Explanation

For 4 degrees of freedom the 5% critical value is 9.49, and the statistic equals this boundary, so it does not exceed it and we fail to reject. A hypothesis test never proves the null hypothesis true.

Question 13Statistics

A random variable X has a uniform distribution on the integers 1 to 6 inclusive. What is the expected value E(X)?

  1. 3.0
  2. 6.0
  3. 4.0
  4. 3.5 Correct answer.

Explanation

For a discrete uniform distribution on 1 to 6, the mean is the average of the endpoints, (1 + 6)/2 = 3.5. Equivalently the sum of all values divided by six gives 21/6 = 3.5.

Question 14Statistics

A researcher increases the sample size from 100 to 400 while keeping the sample standard deviation the same. What happens to the width of a confidence interval for the mean?

  1. It quarters.
  2. It stays the same.
  3. It halves. Correct answer.
  4. It doubles.

Explanation

The standard error is s/√n, and multiplying n by four multiplies √n by two, so the standard error is halved. The interval width is proportional to the standard error, so it also halves.

Question 15Statistics

A bag contains 5 red and 3 blue counters. Two counters are drawn without replacement. What is the probability both are red?

  1. 10/56
  2. 5/8
  3. 5/14 Correct answer.
  4. 25/64

Explanation

The probability the first is red is 5/8, and given that, the second is red with probability 4/7. Multiplying gives 20/56 = 5/14.

Question 16Statistics

A bootstrap distribution of 1000 sample means has a standard deviation of 1.8 and a mean of 25. Which statement about the bootstrap method is correct?

  1. The bootstrap standard deviation estimates the population standard deviation directly.
  2. The bootstrap distribution can only be used when the sample size exceeds 500.
  3. The bootstrap requires the population to be normally distributed.
  4. The bootstrap distribution estimates the sampling distribution of the sample mean. Correct answer.

Explanation

Bootstrap resampling approximates the sampling distribution of a statistic by repeatedly resampling from the observed data. The spread of that bootstrap distribution estimates the standard error of the statistic.

Question 17Statistics

A time-series plot shows a clear increasing trend plus regular seasonal peaks every December. Which forecasting approach is most appropriate?

  1. Exponential smoothing with no seasonal component
  2. Simple linear regression on time alone
  3. Decomposition into trend and seasonal components Correct answer.
  4. Calculating the mean of the entire series

Explanation

When a series contains both trend and seasonality, decomposition separates the trend, seasonal and residual components so each can be modelled and recombined. Ignoring either component produces biased forecasts.

Question 18Statistics

Which statement correctly distinguishes an observational study from an experiment?

  1. An observational study can establish causation if the sample is random.
  2. An observational study cannot use random sampling.
  3. An experiment always uses a larger sample than an observational study.
  4. An experiment assigns treatments to units, allowing causal conclusions. Correct answer.

Explanation

In an experiment the researcher controls treatment assignment, so randomisation can support causal claims. Observational studies merely observe units without assigning treatments, so causation is harder to establish.

Question 19Statistics

A test statistic for a one-sample t-test is t = 2.1 with 15 degrees of freedom. Which feature of the t-distribution is used when finding the p-value?

  1. Its mean equal to the degrees of freedom
  2. Its symmetry about the value 15
  3. Its uniform shape across all degrees of freedom
  4. Its heavier tails compared with the standard normal distribution Correct answer.

Explanation

The t-distribution is symmetric about zero with heavier tails than the normal, especially for small degrees of freedom. This affects the critical values and p-values used in the test.

Question 20Statistics

A simulation is used to estimate a probability by running 10,000 trials. Which change would most reduce the variability of the estimated probability?

  1. Changing the simulation model between runs
  2. Repeating the same 10,000 trials with the same random seed
  3. Increasing the number of trials to 40,000 Correct answer.
  4. Reporting the median of the trial outcomes

Explanation

Simulation estimates have standard error proportional to 1/√N, so quadrupling the number of trials halves the standard error. More independent trials reduce the variability of the estimate.

Question 21Statistics

Two independent random variables X and Y have Var(X) = 4 and Var(Y) = 9. What is Var(X + Y)?

  1. 5
  2. 13 Correct answer.
  3. 36
  4. 6

Explanation

For independent random variables the variance of a sum is the sum of the variances, so Var(X + Y) = 4 + 9 = 13. Independence removes any covariance term.

Question 22Statistics

A dataset has an outlier far above the rest. Which measure of centre is most resistant to this outlier?

  1. Mean
  2. Median Correct answer.
  3. Range
  4. Standard deviation

Explanation

The median depends on the position of the middle value, so a single extreme observation does not shift it much. The mean, range and standard deviation are all sensitive to extreme values.

Question 23Statistics

A researcher tests whether the mean score differs between three teaching methods using a one-way ANOVA. What is the null hypothesis?

  1. The three population variances are equal.
  2. At least two of the sample means are equal.
  3. All three population means are equal. Correct answer.
  4. The overall mean score is zero.

Explanation

A one-way ANOVA tests whether the population means of the groups are all equal against the alternative that at least one differs. The variances being equal is an assumption, not the hypothesis.

Question 24Statistics

A random variable X has P(X = 1) = 0.3, P(X = 2) = 0.5 and P(X = 3) = 0.2. What is E(X)?

  1. 1.9 Correct answer.
  2. 6.0
  3. 1.5
  4. 2.0

Explanation

The expectation is the sum of each value times its probability: 1(0.3) + 2(0.5) + 3(0.2) = 0.3 + 1.0 + 0.6 = 1.9.

Question 25Statistics

A survey asks 500 randomly selected voters their preferred party. Which condition is most important for the sample proportion to be approximately normally distributed?

  1. The respondents must all answer honestly.
  2. The population must be less than 1000 voters.
  3. Both np and n(1 − p) should be reasonably large. Correct answer.
  4. The sample must be exactly 10% of the population.

Explanation

The sampling distribution of a proportion is approximately normal when the expected numbers of successes and failures, np and n(1 − p), are both sufficiently large. Honest responses affect validity but not the distributional approximation.

Question 26Statistics

In a study of the relationship between daily screen time and sleep duration, the least-squares regression line is sleep = 8.2 - 0.35 × screen, where sleep is in hours and screen is in hours. Which interpretation of the slope is correct?

  1. For each additional hour of sleep, predicted screen time decreases by 0.35 hours.
  2. The correlation between screen time and sleep is -0.35.
  3. When screen time is 0 hours, predicted sleep is 0.35 hours.
  4. For each additional hour of screen time, predicted sleep decreases by 0.35 hours. Correct answer.

Explanation

The slope of a least-squares line gives the predicted change in the response variable for each one-unit increase in the explanatory variable. Here screen time is explanatory and sleep is the response, so each extra hour of screen time lowers predicted sleep by 0.35 hours.

Question 27Statistics

A random variable X has the probability distribution P(X=0)=0.1, P(X=1)=0.4, P(X=2)=0.3, P(X=3)=0.2. What is Var(X)?

  1. 2.10
  2. 1.00
  3. 1.80
  4. 0.81 Correct answer.

Explanation

First find E(X) = 0(0.1)+1(0.4)+2(0.3)+3(0.2) = 1.6. Then E(X²) = 0(0.1)+1(0.4)+4(0.3)+9(0.2) = 3.4, so Var(X) = 3.4 - 1.6² = 3.4 - 2.56 = 0.84... check: 3.4 - 2.56 = 0.84. Recomputing: E(X) = 0.4+0.6+0.6 = 1.6. E(X²) = 0.4+1.2+1.8 = 3.4. Var = 3.4 - 2.56 = 0.84.

Question 28Statistics

A 90% confidence interval for a population mean is calculated as (12.4, 15.6). Which statement about this interval is correct?

  1. There is a 90% probability that the true mean lies between 12.4 and 15.6.
  2. In repeated sampling, about 90% of intervals built this way would contain the true mean. Correct answer.
  3. About 90% of the sample values lie between 12.4 and 15.6.
  4. If the sample were repeated, 90% of new sample means would fall in this interval.

Explanation

The confidence level describes the long-run capture rate of the procedure: about 90% of intervals constructed this way contain the true population mean. Once computed, a specific interval either contains the mean or does not.

Question 29Statistics

A one-sample t-test for a mean gives t = 2.45 with 12 degrees of freedom, testing H0: μ = 30 against Ha: μ ≠ 30. Using a two-tailed test, which conclusion is appropriate at the 5% significance level?

  1. Reject H0 because t exceeds 1.96.
  2. Fail to reject H0 because the p-value is greater than 0.05.
  3. Reject H0 because the p-value is less than 0.05. Correct answer.
  4. Accept H0 because 2.45 is small.

Explanation

For 12 degrees of freedom the two-tailed 5% critical value is about 2.179. Since 2.45 exceeds 2.179, the p-value is below 0.05 and H0 is rejected.

Question 30Statistics

A scatterplot of two variables shows a strong curved (U-shaped) relationship. Which statement about using Pearson's correlation r is correct?

  1. r will be close to -1 because the curve turns downward.
  2. r is invalid only when one variable is categorical.
  3. r will be close to 1 because the relationship is strong.
  4. r may be near zero even though the variables are strongly related. Correct answer.

Explanation

Pearson's r measures the strength of a linear relationship only. A strong but non-linear relationship can produce an r near zero, which is why the scatterplot should be checked first.

Question 31Statistics

A large population has proportion p = 0.30 of successes. Random samples of size n = 200 are taken. What is the standard deviation of the sample proportion p̂?

  1. 0.00105
  2. 0.0324 Correct answer.
  3. 0.2100
  4. 0.0458

Explanation

The standard deviation of p̂ is sqrt(p(1-p)/n) = sqrt(0.30 × 0.70 / 200) = sqrt(0.00105) ≈ 0.0324. The success/failure conditions are met because np = 60 and n(1-p) = 140 are both at least 10.

Question 32Statistics

A manager wants to estimate the mean waiting time at a call centre. Which change would produce the narrowest confidence interval, all else being equal?

  1. Decreasing the sample size and lowering the confidence level
  2. Increasing the sample size and raising the confidence level
  3. Decreasing the sample size and raising the confidence level
  4. Increasing the sample size and lowering the confidence level Correct answer.

Explanation

Interval width grows with the critical value and shrinks with the standard error. A larger sample reduces the standard error, and a lower confidence level reduces the critical value, so both changes narrow the interval.

Question 33Statistics

Two events A and B are independent with P(A) = 0.5 and P(B) = 0.4. What is P(A | B)?

  1. 0.5 Correct answer.
  2. 0.9
  3. 0.2
  4. 0.8

Explanation

For independent events, the conditional probability P(A | B) equals the unconditional probability P(A). Since A and B are independent, knowing B occurred does not change the probability of A, so P(A | B) = 0.5.

Question 34Statistics

A political pollster surveys 1000 randomly selected voters and finds 54% support a candidate. A 95% confidence interval is (0.509, 0.571). Which statement about the margin of error is correct?

  1. The margin of error is about 0.031, half the width of the interval. Correct answer.
  2. The margin of error cannot be found without the population proportion.
  3. The margin of error is about 0.062, the full width of the interval.
  4. The margin of error is 0.540, the sample proportion.

Explanation

A confidence interval has the form estimate ± margin of error, so the margin of error is half the interval width. The width is 0.571 - 0.509 = 0.062, so the margin of error is about 0.031.

Question 35Statistics

A 2×3 two-way table of counts is analysed with a chi-square test for independence. How many degrees of freedom does the test have?

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

Explanation

For a chi-square test for independence in a table with r rows and c columns, the degrees of freedom are (r-1)(c-1). Here (2-1)(3-1) = 1 × 2 = 2.

Question 36Statistics

A researcher wants to use a normal approximation for the distribution of the sample proportion p̂. Which condition must be checked?

  1. The sample proportion is exactly 0.5.
  2. The population size is less than 10 times the sample size.
  3. Both np and n(1-p) are at least 10. Correct answer.
  4. The sample size is at least 30.

Explanation

The normal approximation to the sampling distribution of p̂ is reasonable when the expected numbers of successes and failures are both large enough, commonly np ≥ 10 and n(1-p) ≥ 10.

Question 37Statistics

A dataset contains the values 3, 5, 7, 8, 9, 10, 11, 12, 14, 60. Which measure best describes the typical value without being distorted by the value 60?

  1. Median Correct answer.
  2. Range
  3. Standard deviation
  4. Mean

Explanation

The median depends on the position of the middle value, so an extreme observation such as 60 does not greatly affect it. The mean, range, and standard deviation are all sensitive to extreme values.

Question 38Statistics

A random variable X has E(X) = 10 and Var(X) = 4. A new variable is defined as Y = 3X - 5. What is Var(Y)?

  1. 12
  2. 7
  3. 31
  4. 36 Correct answer.

Explanation

For Y = aX + b, the variance is Var(Y) = a²Var(X). Adding or subtracting a constant does not change the variance, so Var(Y) = 3² × 4 = 36.

Question 39Statistics

A simulation is used to estimate the probability of a rare event, and the estimated probability is 0.012 based on 5000 trials. Which statement about the simulation is most appropriate?

  1. The estimate has noticeable simulation error, so more trials would improve precision. Correct answer.
  2. The estimate is biased because the event is rare.
  3. The estimate cannot be used because the probability is below 0.05.
  4. The estimate is exact because the simulation was run many times.

Explanation

Simulation estimates carry random error that decreases as the number of trials increases. With only about 60 occurrences of a rare event, the estimate is imprecise, so running more trials would improve it.

Question 40Statistics

An experiment randomly assigns 60 volunteers to either a new supplement or a placebo and compares weight loss. Which feature of this design supports a causal conclusion?

  1. Measuring weight loss after the treatment
  2. Random assignment to treatment groups Correct answer.
  3. Comparing the two groups' average weight loss
  4. Using volunteers who chose to participate

Explanation

Random assignment balances both known and unknown confounding variables across groups, which allows a causal interpretation of a difference between groups. Merely comparing groups or measuring an outcome does not remove confounding.

Question 41Statistics

A random sample of 16 observations from a normal population gives a sample mean of 42 and sample standard deviation of 8. A 95% confidence interval for the mean uses t* = 2.131. What is the interval?

  1. (38.08, 45.92)
  2. (40.00, 44.00)
  3. (34.16, 49.84)
  4. (37.74, 46.26) Correct answer.

Explanation

The standard error is s/√n = 8/4 = 2. The margin of error is t* × SE = 2.131 × 2 = 4.262, so the interval is 42 ± 4.262, giving (37.74, 46.26).

Question 42Statistics

In a hypothesis test, the p-value is 0.08. Which interpretation of this p-value is correct?

  1. The probability that the alternative hypothesis is true is 0.08.
  2. There is an 8% chance of making a Type I error in this test.
  3. Assuming H0 is true, there is an 8% chance of a test statistic at least as extreme as observed. Correct answer.
  4. There is an 8% chance that H0 is true.

Explanation

A p-value is the probability, computed assuming H0 is true, of obtaining a test statistic at least as extreme as the one observed. It is not the probability that H0 is true nor a direct error probability.

Question 43Statistics

A time series of monthly ice-cream sales shows a strong seasonal pattern with higher sales in summer. Which forecasting method is most appropriate?

  1. A simple linear trend line only
  2. A moving average with a very short window only
  3. Using the most recent month's sales as the forecast
  4. Seasonal decomposition or a seasonal model Correct answer.

Explanation

When a series has a clear repeating seasonal pattern, a method that explicitly models the seasonal component (such as seasonal decomposition) captures the pattern and produces better forecasts than methods that ignore it.

Question 44Statistics

A researcher compares mean test scores across four teaching methods using a one-way ANOVA. The ANOVA F test is significant. What does this result establish?

  1. At least one pair of group means differs. Correct answer.
  2. All four group means differ from each other.
  3. The sample means are all equal to the population mean.
  4. The variances of the four groups are equal.

Explanation

A significant one-way ANOVA F test indicates that not all population means are equal, meaning at least one pair differs. It does not identify which pairs differ; a post-hoc procedure is needed for that.

Question 45Statistics

A bag contains 4 green and 6 yellow marbles. One marble is drawn and not replaced, then a second is drawn. What is the probability that both marbles are green?

  1. 8/45
  2. 2/15 Correct answer.
  3. 1/5
  4. 4/25

Explanation

The probability the first is green is 4/10. Given that, 3 green remain among 9 marbles, so the second is green with probability 3/9. Thus P(both green) = (4/10)(3/9) = 12/90 = 2/15.

Question 46Statistics

A bootstrap distribution of the difference between two sample means has a mean of 3.2 and a standard deviation of 0.9. Which statement about a 95% bootstrap confidence interval is correct?

  1. It must contain the value 0.
  2. It is approximately 3.2 ± 1.96 × 0.9, giving roughly (1.44, 4.96). Correct answer.
  3. It is exactly 3.2 ± 0.9, giving (2.3, 4.1).
  4. It cannot be computed without the original data.

Explanation

When the bootstrap distribution is approximately normal, a 95% interval can be formed as estimate ± 1.96 × bootstrap standard error. Here that gives 3.2 ± 1.764, or roughly (1.44, 4.96).

Question 47Statistics

A study reports a correlation of r = 0.62 between hours of study and exam score, based on 50 students. A 95% confidence interval for the population correlation is (0.41, 0.77). Which interpretation is correct?

  1. 95% of students have correlations between 0.41 and 0.77.
  2. There is a 95% probability that r = 0.62.
  3. The relationship is definitely non-linear.
  4. The data are consistent with a moderate to strong positive linear association in the population. Correct answer.

Explanation

The interval estimates the population correlation and lies entirely positive, indicating a moderate to strong positive linear association is plausible. The width reflects uncertainty from the sample size of 50.

Question 48Statistics

A game charges $2 to play. A player wins $10 with probability 0.1 and wins nothing otherwise. What is the expected net gain per game?

  1. $0.80
  2. $1.00
  3. -$1.00 Correct answer.
  4. -$0.80

Explanation

The expected winnings are 0.1 × $10 = $1.00. Subtracting the $2 cost gives an expected net gain of $1.00 - $2.00 = -$1.00 per game.

Question 49Statistics

A researcher wants to estimate the proportion of students who own a car, with a margin of error of about 3% at 95% confidence. Using the conservative estimate p = 0.5, what sample size is needed?

  1. 384
  2. 2401
  3. 1068 Correct answer.
  4. 600

Explanation

Using n = (z*/m)² p(1-p) with z* = 1.96, m = 0.03 and p = 0.5 gives n = (1.96/0.03)² × 0.25 ≈ 1067.1, which rounds up to 1068.

Question 50Statistics

A student tests H0: μ = 50 against Ha: μ > 50 using a sample of size 25 from a normal population with unknown standard deviation. Which test is appropriate?

  1. A chi-square goodness-of-fit test
  2. A one-sample t-test with 24 degrees of freedom Correct answer.
  3. A one-sample z-test using the normal distribution
  4. A two-sample t-test

Explanation

When the population standard deviation is unknown and the sample is small, the sample standard deviation is used and the test statistic follows a t-distribution with n-1 = 24 degrees of freedom.