A) The hypotheses for the test in this scenario can be stated as follows:
Null hypothesis (H0): The new material does not increase the mileage of the tires produced.
Alternative hypothesis (Ha): The new material increases the mileage of the tires produced by at least 10%.
B) To write a rejection rule for the test with a significance level (alpha) of 0.05, we need to determine the critical value.
Since the standard deviation (σ) of the average life of your current tires is known, and you will be testing whether the new material increases the mileage by at least 10%, you can use a one-sample Z-test.
To calculate the critical value, you can use a Z-distribution table or a statistical software. The critical value corresponding to a significance level of 0.05 (α = 0.05) is approximately 1.96 for a two-tailed test.
Rejection Rule: If the calculated test statistic (Z-score) is greater than 1.96 or less than -1.96, you will reject the null hypothesis and conclude that the new material has a significant impact on increasing the mileage of the tires produced. Otherwise, if the calculated test statistic falls within the range of -1.96 to 1.96, you will fail to reject the null hypothesis.