Interpreting a Confidence Interval from a Small Sample
Assume independent packet masses from an approximately normal population. A student measures the mass of 9 randomly selected packets of seeds from a production line and obtains a mean of 50.2 g with a sample standard deviation of 1.5 g. The student constructs a 95% confidence interval for the population mean using the normal critical value 1.96 and reports 50.2 ± 0.98, i.e., (49.22, 51.18). A classmate says the interval is wrong because the sample is small and the population standard deviation is unknown. The classmate suggests using the t-distribution with 8 degrees of freedom and a critical value of 2.306. The student objects that 'the sample mean is 50.2, so there is a 95% chance the true mean is between 49.22 and 51.18.' The teacher asks the class to resolve all three issues: the correct interval, the degrees of freedom, and the correct interpretation.
1. Compute the correct 95% confidence interval and explain the choice of critical value and degrees of freedom.
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The population standard deviation is unknown, so we use the sample standard deviation and the t-distribution. The standard error is s/√n = 1.5/√9 = 1.5/3 = 0.5 g. Degrees of freedom are n - 1 = 8. The critical t-value for 95% confidence with 8 df is about 2.306. The margin of error is 2.306 × 0.5 = 1.153 g. The interval is 50.2 ± 1.153, i.e., approximately (49.05, 51.35) g. The student's normal-based interval used 1.96 × 0.5 = 0.98, which is too narrow because it ignores the extra uncertainty from estimating σ with s. Using z when σ is unknown and n is small understates the margin of error. The classmate is correct: with n = 9 and unknown σ, the t-distribution with 8 df is appropriate. The t critical value is larger than z, widening the interval to reflect the added uncertainty.
2. Evaluate the student's interpretation: 'there is a 95% chance the true mean is between 49.22 and 51.18.'
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The interpretation is a common but incorrect statement. In frequentist inference, the population mean is a fixed (unknown) constant, not a random variable, so it does not have a probability of lying in a particular realised interval. The 95% refers to the procedure: if we repeated the sampling many times and constructed an interval each time, about 95% of those intervals would contain the true mean. For the single interval we computed, we can say we are 95% confident in the method, not that there is a 95% probability the parameter is inside this specific interval. A correct phrasing: 'We are 95% confident that the true mean packet mass lies between 49.05 g and 51.35 g,' where 'confident' refers to the long-run coverage of the procedure. The student's statement also uses the wrong interval (49.22, 51.18), which came from the z-based calculation.