Question

Old MacDonald, with 1500 swine, had raised the largest pen of pigs in Pittsburg, Pennsylvania. He began using a custom blend of nutritional goulash. A random sample of 37 pigs was obtained and their protein level was measured revealing an average increase of 6.7 daltons. Prior to the introduction of the custom blend of goulash, the variance of protein level for the whole pen was 2.7 daltons, and the average protein level was 5.2 daltons.

Use an appropriate hypothesis test with a level of significance at 7% to determine if there is enough evidence to support a claim that Old MacDonald’s custom blend of nutritional goulash could increase protein levels above 5.2 daltons. If you use a t-test, use df = 3

Answer 1

To test if Old MacDonald's goulash increases pig protein levels, a one-sample t-test is used with H₀: μ = 5.2 daltons and H₁: μ > 5.2 daltons. If the calculated t is greater than the critical t value for df=36 at 7% significance, H₀ is rejected supporting the claim of increased levels. DF should be sample size minus 1, not 3 as indicated unless it's a specific question requirement.

Step-by-step explanation:

To determine if Old MacDonald’s custom blend of nutritional goulash could significantly increase protein levels in pigs, we can use a one-sample t-test. We will test the null hypothesis (H₀) that there is no increase in protein level (population mean μ = 5.2 daltons) against the alternative hypothesis (H₁) that there is an increase (population mean μ > 5.2 daltons).

Given that the sample mean is 6.7 daltons, the population variance is 2.7 daltons, and the sample size is 37 pigs, we can calculate the t statistic as follows:

T = (Sample mean − Population mean) / (Standard deviation of the sample / √Sample size)

Since the population standard deviation is not provided, we use the sample standard deviation which is the square root of the variance. The t statistic can be calculated and then compared to the t critical value for df = 36 (not 3, as the df is generally the sample size minus 1) at the 7% significance level.

If the calculated t statistic is greater than the critical t value, we reject the null hypothesis and conclude that there is enough evidence to support the claim that the goulash increases protein levels. If the calculated t statistic does not exceed the critical t value, we fail to reject the null hypothesis.

We must pay attention to the degrees of freedom; in this question, df should be the sample size minus 1 (37-1=36), not 3 as indicated. However, if we follow the question's prompt using df=3, then it might be an error or a specific requirement of the question. The t critical value would then be obtained from a t distribution table or software with the specified df.