(Solution) - A shoe manufacturer claims that athletes can decrease their times -(2025 Original AI-Free Solution)

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A shoe manufacturer claims that athletes can decrease their times in the 40-yard dash using the manufacturer?s training shoes. The 40-yard dash times of 12 randomly selected athletes are measured. After the athletes have used the shoes for 8 months, their 40-yard dash times are measured again. The times (in seconds) are listed below. At ? = 0.05, is there enough evidence to support the manufacturer?s claim? Assume the times are normally distributed.
Before After
4.85 ???. 4.78
4.90 ???. 4.90
5.08 ???. 5.05
4.72 ???. 4.65
4.62 ???. 4.64
4.54 ???. 4.50
5.25 ???. 5.24
5.18 ???. 5.27
4.81 ???. 4.75
4.57 ???. 4.43
4.63 ???. 4.61
4.77 ???. 4.82
a. Identify the claim and state H0 and Ha.
b. Identify the level of significance ? and the degrees of freedom.
c. Find the critical value and identify the rejection region.
d. Calculate  and sd.
e. Find the standardized test statistic t. Sketch a graph.
f. Decide whether to reject the null hypothesis.
g. Interpret the decision in the context of the original claim.