A friend claims he scores 1400 on a game; you collect a random sample of 12 scores; is there evidence that the population mean is less than 1400?

A friend claims he scores 1400 on a game; you collect a random sample of 12 scores; is there evidence that the population mean is less than 1400?

• A. Two-Sample T-Test
• B. One-Sample T-Test
• C. One-Proportion Z-Test
• D. Matched-Pairs T-Test

Answer: (B)

Testing whether a single population mean is below a claimed value based on a small random sample with an unknown population standard deviation calls for a one-sample t-test. You’re evaluating if mu, the true mean, is less than 1400 using data from one sample of 12 scores. Set up hypotheses: the null says the mean is 1400, while the alternative claims the mean is smaller than 1400. Because the population standard deviation isn’t known and the sample size is small, you use the t distribution with degrees of freedom equal to n minus 1 (11). The test statistic is t = (x̄ − 1400) / (s / sqrt(n)), where x̄ is the sample mean and s is the sample standard deviation. You then compare that t value to the appropriate one-sided critical value, or compute a p-value for the lower tail. If the result is statistically significant, you have evidence that the population mean is less than 1400. Why not the other tests: a two-sample t-test compares means from two independent groups, which isn’t the situation here. a one-proportion z-test is for categorical data analyzed as proportions, not means. a matched-pairs t-test is for paired observations (such as before-and-after measurements on the same subjects), not a single random sample meant to reflect a population mean.

Testing whether a single population mean is below a claimed value based on a small random sample with an unknown population standard deviation calls for a one-sample t-test. You’re evaluating if mu, the true mean, is less than 1400 using data from one sample of 12 scores.

Set up hypotheses: the null says the mean is 1400, while the alternative claims the mean is smaller than 1400. Because the population standard deviation isn’t known and the sample size is small, you use the t distribution with degrees of freedom equal to n minus 1 (11). The test statistic is t = (x̄ − 1400) / (s / sqrt(n)), where x̄ is the sample mean and s is the sample standard deviation. You then compare that t value to the appropriate one-sided critical value, or compute a p-value for the lower tail. If the result is statistically significant, you have evidence that the population mean is less than 1400.

Why not the other tests: a two-sample t-test compares means from two independent groups, which isn’t the situation here. a one-proportion z-test is for categorical data analyzed as proportions, not means. a matched-pairs t-test is for paired observations (such as before-and-after measurements on the same subjects), not a single random sample meant to reflect a population mean.