Studies conducted by the manufacturer of two different brands of asphalt shingles have shown product weight to be a major factor in the customer's perception of quality. The accompanying table shows the weight (in pounds) from a sample of 75 pallets of brand A shingles and 70 pallets of brand B shingles. Complete parts (a) through (e) below.
a. For the brand A shingles, is there evidence at the 0.01 level of significance that the population mean weight is different from 3,170 pounds? Determine the null hypothesis, H0, and the alternative hypothesis, H1.
H0: μ = 3,170
H1: μ ≠ 3,170
Determine the test statistic. tSTAT = (sample mean - population mean) / (sample standard deviation / √sample size)
Determine the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
State the conclusion. Reject or do not reject H0. There is or is no evidence that the population mean weight is equal to, not equal to, less than, or greater than 3,170 pounds.
b. Interpret the meaning of the p-value in (a). Select the correct choice below and fill in the answer box to complete your choice.
A. If the population mean weight is in fact not equal to 3,170 pounds, there is a [p-value]% chance of observing a sample of 75 pallets that will yield a test statistic more extreme than the test statistic for this sample.
B. If the population mean weight is in fact 3,170 pounds, there is a [p-value]% chance of observing a sample of 75 pallets that will yield a test statistic more extreme than the test statistic for this sample.
C. There is a [p-value]% chance that the alternative hypothesis is true.
D. There is a [p-value]% chance that the null hypothesis is true.
c. For the brand B shingles, is there evidence at the 0.01 level of significance that the population mean weight is different from 3,700 pounds? Determine the null hypothesis, H0, and the alternative hypothesis, H1.
H0: μ = 3,700
H1: μ ≠ 3,700
Determine the test statistic. tSTAT = (sample mean - population mean) / (sample standard deviation / √sample size)
Determine the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
State the conclusion. Reject or do not reject H0. There is or is no evidence that the population mean weight is less than, not equal to, or greater than 3,700 pounds.
d. Interpret the meaning of the p-value in (b). Select the correct choice below and fill in the answer box to complete your choice.
A. There is a [p-value]% chance that the alternative hypothesis is true.
B. If the population mean weight is in fact 3,700 pounds, there is a [p-value]% chance of observing a sample of 70 pallets that will yield a test statistic more extreme than the test statistic for this sample.
C. If the population mean weight is in fact not equal to 3,700 pounds, there is a [p-value]% chance of observing a sample of 70 pallets that will yield a test statistic more extreme than the test statistic for this sample.
D. There is a [p-value]% chance that the null hypothesis is true.
e. In (a) through (d), do you have to be concerned with the normality assumption? Explain.
A. Yes, since the sample sizes are not large enough to enable the test statistic to follow the t distribution, and since it cannot be assumed that the underlying populations are approximately normally distributed.
B. No, since the sample sizes are large enough to enable the test statistic to follow the t distribution.
C. Yes, since the sample sizes are not large enough to enable the test statistic to follow the t distribution regardless of the distributions of the underlying populations.
D. No, since it can be assumed that the underlying populations are approximately normally distributed.