Final answer:
In the context of hypothesis testing in statistics, the question pertains to testing the claim that the standard deviation of hardness indexes for bolts is greater than 28 using a chi-square test with a sample standard deviation of 38 and 15 samples.
Step-by-step explanation:
The subject of this question is Mathematics, specifically hypothesis testing in statistics. We are testing the claim that the standard deviation of the hardness indexes for all bolts is greater than 28 at a 0.10 level of significance. To do this:
- The null hypothesis (H0) is that the population standard deviation is 28: σ = 28.
- The alternative hypothesis (Ha) is that the population standard deviation is greater than 28: σ > 28.
- We have 15 samples, so the degrees of freedom (df) will be 15 - 1 = 14.
- To find the test statistic, we use the formula: χ² = (n - 1)s²/σ², where n is the sample size, s is the sample standard deviation, and σ is the hypothesized population standard deviation. Here, the test statistic would be χ² = (15 - 1)(38)^2/28^2.
Graph representation and p-value calculations would follow, using the appropriate chi-square distribution to find the area to the right of the calculated test statistic value.
Depending on the p-value and our alpha level (0.10), we will then either reject or fail to reject the null hypothesis.
If the p-value is less than 0.10, we reject the null hypothesis and conclude that there is enough evidence to support the claim that the standard deviation is greater than 28.
If the p-value is greater, we do not reject the null hypothesis.