Question

Suppose you own a restaurant and have a cook whose ability and attitude you are suspicious of. One of the dishes on the menu is duck cassoulet, which uses duck legs that have been slow fried over a couple of hours in oil that does not exceed a temperature of 175 degrees. This is a time consuming and monotonous process, but one that results in excellent meat that you sell for a large mark-up. You suspect your cook is lazy and doesn't properly monitor and maintain the oil temperature. You take a random sample of 15 duck legs and take them to a forensics lab where you are able to discover the maximum temperature the meat has reached. Within your sample the mean maximum temperature of the duck legs is 180 degrees with a standard deviation of 4 degrees. Meat cooked precisely to 175 degrees is what your cook is supposed to do.

A) Which of the following is true about a hypothesis test for the claim that your employee is capable (meaning he doesn't over-fry the meat) at the 90% confidence level?
Group of answer choices:
O Reject the null with a test statistic value of 1.83
O Reject the null with a test statistic value of 2.17
O Fail to reject the null with a test statistic value of 1.59
O Fail to the null with a test statistic value of 1.47
O None of the above are true

Answer 1

`t=(180-175)/((4)/(√(15)))=4.84`

`df=n-1=15-1=14`

`p_v =P(t_((14))>4.84)=0.000131`

We got a very low value for the p value so then we have enough evidence to reject the null hypothesis at any significance level commonly used. And the best conclusion based on the possible options are:

None of the above are true

Explanation:

Data given

`\bar X=180` represent the sample mean

`s=4` represent the sample standard deviation

`n=15` sample size

`\mu_o =175` represent the value that we want to test

`\alpha=0.1` represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

System of hypothesis

We need to conduct a hypothesis in order to check if the true mean is above the limit of 175 degrees, the system of hypothesis would be:

Null hypothesis:
`\mu \leq 175`

Alternative hypothesis:
`\mu > 175`

The statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

And replacing we got:

We can replace in formula (1) the info given like this:

`t=(180-175)/((4)/(√(15)))=4.84`

P-value

The degrees of freedom are given by:

`df=n-1=15-1=14`

Since is a one sided test the p value would be:

`p_v =P(t_((14))>4.84)=0.000131`

We got a very low value for the p value so then we have enough evidence to reject the null hypothesis at any significance level commonly used. And the best conclusion based on the possible options are:

None of the above are true