Question Type 3: Performing a hypothesis test comparing two means of unpaired samples with equal variances Exercises

At the 1% significance level, test whether there is a difference in the mean lifetime of two battery brands, A and B, based on the following sample data (measured in hours):

Brand A: $200, 210, 190, 205, 215, 195, 220$ ($n=7$) Brand B: $180, 175, 185, 170, 190, 200, 195, 205$ ($n=8$)

Assume that the lifetimes of both brands follow normal distributions with equal variances.

Ten students took a test before and after a review session. Their scores are shown in the following table:

Test the claim, at the 1% significance level, that the review session improved the students' scores.

Eight pairs of athletes had their running times, in seconds, measured on a grass track and a synthetic track. The results are recorded in the table below.

At the 5% significance level, test the claim that the synthetic track yields faster running times. You may assume that the differences in running times are normally distributed.

A group of 10 people each watched a horror film and a romance film. Their calorie consumption (horror, romance) were recorded as follows:

$(345, 453), (234, 112), (45, 887), (216, 593), (543, 953), (143, 438), (576, 299), (396, 548), (222, 671), (341, 554)$.

At the 5% level of significance, test whether the mean calorie consumption differs between the two film genres.

Two machines produce metal rods. Sample lengths (mm) are recorded as follows:

Machine 1 ($n=10$): $100.5, 101.0, 99.8, 100.2, 100.9, 101.1, 100.4, 100.0, 99.9, 100.7$

Machine 2 ($n=12$): $100.1, 100.3, 99.7, 100.2, 100.5, 99.8, 100.0, 100.4, 100.2, 99.9, 100.6, 100.8$

Test at the 5% significance level whether the mean lengths of the rods produced by the two machines differ, assuming the populations are normally distributed with equal variances.

Seven patients had their blood pressure measured before and after taking a new drug. The results, given as pairs of (before, after) in kPa, are:

$(8.1, 7.5), (7.8, 7.7), (8.5, 8.0), (7.9, 7.4), (8.2, 7.9), (8.0, 7.6), (7.7, 7.2)$.

At the 5% significance level, test whether the drug significantly lowers blood pressure.

Two independent samples measure a new process yield (in %):

Sample A ($n=5$): $12, 15, 14, 10, 13$

Sample B ($n=6$): $16, 18, 17, 19, 20, 22$

Assuming equal variances, test at the 5% level of significance whether the mean yields differ.

A factory measures product weight before and after calibration on 6 items (g): $(50.2, 50.1), (49.8, 49.7), (50.5, 50.4), (50.0, 49.9), (50.3, 50.1), (49.9, 49.8)$

At the $5\%$ level of significance, test whether calibration changes the mean weight.

Two cycling training programs are evaluated based on the mean speeds (km/h) of their participants. The following results were recorded:

Program A: $30, 32, 31, 29, 33, 34, 30, 31, 32$ Program B: $28, 29, 27, 30, 29, 28, 31, 27, 30$

Assuming the populations have equal variances, perform a $t$-test at the 1% significance level to determine whether there is a significant difference between the mean speeds of the two programs.

Perform a hypothesis test at the 5% significance level to determine if the mean calorie consumption differs between viewers of horror and romance films. The horror-film sample (in calories) is: $345, 234, 45, 216, 543, 143, 576, 396$.

The romance-film sample is: $453, 112, 887, 593, 953, 438, 299, 548, 671$.

Assume the two populations are normally distributed and have equal variances.