常態分布與 t 分布的統計推斷：練習題

A researcher wants to test if the average reaction time to a stimulus is significantly different from 0.5 seconds. A random sample of 12 participants yields an average reaction time of 0.53 seconds with a sample standard deviation of 0.08 seconds. Assuming reaction times are normally distributed, what is the value of the test statistic for testing the hypothesis $H_0: \mu = 0.5$ against $H_1: \mu \neq 0.5$?

Using the data from the previous question (Method A: $n_A = 10, \bar{x}_A = 78, s_A = 8$; Method B: $n_B = 12, \bar{x}_B = 72, s_B = 10$), calculate a 95% confidence interval for the difference in population means ($\mu_A - \mu_B$), assuming normal distributions and equal population variances.

A new training program is introduced for factory workers. The productivity of 8 randomly selected workers is measured before and after the program. The differences (After - Before) in productivity scores are: 3, 1, -2, 4, 0, 2, 5, 1. Test whether the training program has improved productivity at the 1% significance level. Assume differences are normally distributed. What is the conclusion?

A company implements a new software package and wants to evaluate its impact on employee task completion time. 12 employees are randomly selected, and their times (in minutes) to complete a standard task are recorded before and after the software implementation. The mean difference (After - Before) was found to be -2.5 minutes, and the standard deviation of these differences was 3.8 minutes. At the 5% significance level, is there evidence that the software has reduced task completion time?

A manufacturing process produces items with a certain measurement. A random sample of 15 items has a mean measurement of 25.4 cm and a sample standard deviation of 1.2 cm. Assuming the measurements are normally distributed, calculate a 90% confidence interval for the population mean measurement.

A random sample of 10 measurements of the breaking strength (in N) of a certain type of fishing line yielded the following results:

45.2, 46.1, 47.3, 45.8, 46.5, 47.0, 46.3, 45.9, 47.1, 46.7.

Assume that the breaking strengths are normally distributed.

Calculate a 95% confidence interval for the true mean breaking strength of this type of fishing line.

A manufacturer claims that the average lifespan of their light bulbs is 1200 hours. A consumer group suspects that the average lifespan is less than 1200 hours. They test a random sample of 8 light bulbs and record their lifespans (in hours) as follows:

1180, 1195, 1150, 1205, 1170, 1165, 1190, 1175.

Assuming the lifespans are normally distributed, carry out a hypothesis test at the 5% significance level to determine if there is evidence to support the consumer group's suspicion.