Question

1 of 2

Geoff is the proud owner of a restaurant. He is interested in determining whether his Wagyu beef or Hiramasa kingfish sashimi should be marketed as the Geoff Special. Geoff has selected a random sample of 20 people to taste his Wagyu beef and give it a score out of 100. He also selected a different random sample of 20 people to taste his Hiramasa kingfish sashimi and give it a score out of 100.

The sample mean score given to the Wagyu beef dish was calculated as 67.95. The sample standard deviation of the scores for the Wagyu beef dish was calculated as 5. The sample mean score given to the Hiramasa kingfish sashimi dish was calculated as 68.65. The sample standard deviation of the scores for the Hiramasa kingfish sashimi dish was calculated as 5. The population standard deviations of the scores for each dish are unknown.

You may find this Student's t distribution table useful throughout the following questions. Note that Geoff always aims to use the easiest possible calculations, and so when using a two-sample t-test he will use the simplified formula for degrees of freedom whenever possible.

a)Geoff would like to test whether the mean scores of each of these dishes are equal. He has constructed a hypothesis test with H₀: μ_W = μ_H and H_A: μ_W ≠ μ_H. Calculate the test statistic (t) for this hypothesis test. Give your answer to 2 decimal places.

t =

b)Using the test statistic for Geoff's hypothesis test and a 95% confidence level, Geoff should accept, reject, not reject the null hypothesis.

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A study compared the individual pre-tax yearly income earned by residents from two states. The following table lists the statistics resulting from this study:

Yearly Income
Location	Sample Size	Sample Mean ($'000s)	Sample Standard Deviation ($'000s)
State A	51	93	17
State B	55	50	37

A part of the study involved calculating the upper and lower bound of the 90% confidence interval of the mean difference (State A - State B) between the income earned by individuals from the two states. For the purposes of this study, the simplified formula for the number of degrees of freedom in the appropriate t-distribution will be used. Calculate the confidence interval. Give your answers to 2 decimal places. You may find this Student's t distribution table useful.

a)Lower bound =

b)Upper bound =

Answer 1

(1a) The test statistics for testing the hypothesis is given by

where n and m are sample sizes sw and sh are sample sd's W,H are sample means and S is pooled variance

(1b)t-critical value at 95% confidence and 38 degrees of freedom(20+20-2) is 1.68 so we see that the test statistic lies between (-1.68,+1.68) so we accept the null hypothesis

(2) The (1-alpha) confidence interval for the difference of two means is given by

Considering the State A as X and State Y as B we can get the above values as

degrees of  freedom and the acceptance region is thus

the interval is thus