Question

Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P value method or the traditional method of testing hypotheses. Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that hte new production method has errors with a standard deviation greater than 32.2 ft which was the standard deviation for the old production method. If it appears that the standard deviation is greater does the new production method appear to be better or worse than the old method?

Answer 1

The hypothesis test involves setting a null hypothesis that the new method's standard deviation is equal to the old one and an alternative hypothesis that it is greater.

Step-by-step explanation:

Testing the Claim Using Hypothesis Testing

When testing the claim of whether the new production method for aircraft altimeters has a standard deviation of errors greater than the old method's 32.2 ft, we begin by defining our null hypothesis (H0) and alternative hypothesis (Ha). For this scenario:

• H0: σ = 32.2 (The standard deviation is equal to that of the old method.)
• Ha: σ > 32.2 (The standard deviation is greater than that of the old method.)

This is a one-tailed test as we are only interested in whether the new method has a greater standard deviation, not if it is less or simply different.

The test statistic for this hypothesis test is typically calculated using the chi-square distribution, which will require the sample variance and the sample size to compute. Once the test statistic is calculated, it can be compared with the critical value from the chi-square distribution table, or the p-value can be calculated to determine the evidence against the null hypothesis.

If the p-value is less than the chosen alpha level of significance (0.05 in this case), we reject the null hypothesis. If the p-value is greater, we do not reject the null hypothesis. A decision to reject the null hypothesis indicates there is sufficient evidence to support the claim that the new method has a greater standard deviation in errors and thus may be considered worse than the old method in terms of consistency.

Moreover, it's important to understand what a Type I error (p-value erroneously indicating to reject a true null hypothesis) and a Type II error (failing to reject a false null hypothesis) mean in this context.

In relation to the question, the calculation of the p-value or comparison with the critical value allows for a conclusion to be drawn. If you've calculated a test statistic of 2.13 and a corresponding p-value of 0.0165 and the alpha is set to 0.05, you would reject the null hypothesis. This indicates that there's sufficient evidence to support the claim that the standard deviation of the new method is greater than 32.2 ft, suggesting that the new production method may be worse than the old one in terms of precision.