Question

WineOrigin In one study, 39 diners were given a free glass of Cabernet Sauvignon wine to accompany a French meal. Although the wine was identical, half the bottle labels claimed the wine was from California and the other half claimed it was from North Dakota. The following table summarizes the grams of entree and wine consumed during the meal. Did the patrons who thought that the wine was from California consume more? Wine Label n x¯ s Entree California 24 499.8 87.2 North Dakota 15 439.0 89.2 Wine California 24 100.8 23.3 North Dakota 15 110.4 9.0 1.) What significance test should be performed? Choice A: Two-sample t test Choice B: One-sample t test Choice C: Paired t test 2.) What are the correct hypotheses to test this claim? Choice A: H0: uCali = uND Ha: uCali < uND Choice B: H0: uCali = uND Ha: uCali > uND Choice C: H0: ud = 0 Ha: ud < 0 Choice D: H0: ud = 0 Ha: ud > 0 3.) Find the test statistics and conservative degrees of freedom for both the amount of entree consumed and the amount of wine consumed. Use 2 decimals for the test statistics. Entree: t-statistic = ?? degrees of freedom = ?? Wine: t-statistic = ?? degrees of freedom = ?? 4.) Which are true regarding the conclusion for the entree consumption data? Select all that apply: 3 are correct. ~ p-value < o ~ p-value > o ~ statistically significant ~ not statistically significant ~ Reject H0 ~ Fail to reject H0 5.) Which are true regarding the conclusion for the wine consumption data? Select all that apply: 3 are correct. ~ p-value < o ~ p-value > o ~ statistically significant ~ not statistically significant ~ Reject H0 ~ Fail to reject H0 6.) A manufacturer wishes to compare the mean lengths of items produced by two metal stamping processes. How many parts should be sampled to estimate the difference in means to within ±0.09mm with 95% confidence? Previous studies indicate the standard deviation for both processes is about 1.25 mm.

Answer 1

Explanation:

1) The significance test that should be performed in this scenario is a Two-sample t test.

2) The correct hypotheses to test this claim are:

- H0: uCali = uND (The mean consumption of patrons who thought the wine was from California is equal to the mean consumption of patrons who thought the wine was from North Dakota)

- Ha: uCali ≠ uND (The mean consumption of patrons who thought the wine was from California is not equal to the mean consumption of patrons who thought the wine was from North Dakota)

3) To find the test statistics and conservative degrees of freedom:

- For the amount of entree consumed:

- t-statistic = (x¯Cali - x¯ND) / sqrt((s^2Cali/nCali) + (s^2ND/nND))

- degrees of freedom = min(nCali - 1, nND - 1)

- For the amount of wine consumed:

- t-statistic = (x¯Cali - x¯ND) / sqrt((s^2Cali/nCali) + (s^2ND/nND))

- degrees of freedom = min(nCali - 1, nND - 1)

4) The true statements regarding the conclusion for the entree consumption data are:

- p-value > α (alpha)

- not statistically significant

- Fail to reject H0

5) The true statements regarding the conclusion for the wine consumption data are:

- p-value < α (alpha)

- statistically significant

- Reject H0

6) To estimate the difference in means to within ±0.09mm with 95% confidence, the sample size needed can be calculated using the formula:

- n = (z^2 * σ^2) / E^2

- where z is the z-value for the desired confidence level (for 95% confidence, z ≈ 1.96), σ is the standard deviation, and E is the desired margin of error.

- In this case, n = (1.96^2 * 1.25^2) / 0.09^2 ≈ 49.84

- Therefore, a sample size of at least 50 parts should be sampled to estimate the difference in means within ±0.09mm with 95% confidence.