Domino's pizza claims that its mean delivery time has a mean of at most 30 minutes. A random selection of 16 delivery times was found to have a mean of 28.5 minutes and a standard deviation of 3.5 minutes.

a) What assumption(s) do you need to conduct a hypothesis testing on this problem?

b) Assume this assumption(s) in (a) is satisfied, and check if there is enough evidence to support their claim at the 0.01 level of significance.

a) Check Explanation

b) The p-value obtained is greater than the significance level at which the test was performed, hence, there isn't enough evidence to support the claim and conclude that the mean delivery time for Domino's Pizza is less than 30 minutes.

Explanation:

a) The assumptions required to conduct a suitable t-test about a mean include

- The data must be obtained from a random sample of the whole population (simple random sample).

- Individual sample data values are expected to be independent of one another.

- The sampling distribution should be normal or approximately normal.

b) For hypothesis testing, the first thing to define is the null and alternative hypothesis.

The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.

For this question, we want to investigate if the mean delivery time for the Domino's Pizza is truly less than 30 minutes.

Hence, the null hypothesis is that there isn't enough evidence to conclude that the mean delivery time for Domino's Pizza is less than 30 minutes. That is, the mean delivery time for Domino's Pizza is greater than 30 minutes.

The alternative hypothesis will now be that there is significant evidence to conclude that the mean delivery time for Domino's Pizza is less than 30 minutes.

Mathematically,

The null hypothesis is given as

H₀: μ ≥ 30 minutes

The alternative hypothesis is given as

Hₐ: μ < 30 minutes

To do this test, we will use the t-distribution because no information on the population standard deviation is known

So, we compute the t-test statistic

t = (x - μ)/σₓ

x = sample mean = 28.5 minutes

μ₀ = The standard we are comparing against = 30 minutes

σₓ = standard error = [σ/√n]

σ = Sample standard deviation = 3.5 minutes

where n = Sample size = 16

σₓ = [3.5/√16] = 0.875

t = (28.5 - 30) ÷ 0.875

t = -1.71

checking the tables for the p-value of this t-statistic

Degree of freedom = df = n - 1 = 16 - 1 = 15

Significance level = 0.01

The hypothesis test uses a one-tailed condition because we're testing only in one direction.

p-value (for t = -1.71, at 0.01 significance level, df = 15, with a one tailed condition) = 0.053932

The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

So, for this question, significance level = 0.01

p-value = 0.053932

0.053932 > 0.01

Hence,

p-value > significance level

This means that we fail to reject the null hypothesis & say that there isn't enough evidence to conclude that the mean delivery time for Domino's Pizza is less than 30 minutes.

Hope this Helps!!!

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