Question

A local retailer claims that the mean waiting time is less than 8 minutes. A random sample of 20 waiting times has a mean of 6.3 minutes with a standard deviation of 2.1 minutes. At α = 0.01, test the retailerʹs claim. Assume the distribution is normally distributed. Use any method, however, follow the PHANTOMS acronym.

P - Parameter Statement

H - Hypotheses

A - Assumptions & Conditions

N - Name the Test and state the curve you're using

T - Test Statistic - Round your value to TWO decimals and state the command you used to find this value

O - Obtain the P-Value or Critical Value . State the command you are using to find these values

M - Make a Decision about the Null Hypothesis and explain why

S - State Your Conclusion About the Claim

Answer 1

Given:

Population mean(µ) = 8 minutes

Sample size(n) = 20

Sample mean(x bar) = 6.3 minutes

Sample standard deviation (s) = 2.1 minutes

α = 0.01

1) Parameter statement:

The claim statement is, "Mean waiting time is less than 8 minutes".

That is, the claim statement goes under the alternative hypothesis.

The population parameter is, µ.

Hence, alternative hypothesis H1: µ < 8.

2) Hypothesis

The null hypothesis and alternative hypothesis are opposite to each other.

Null hypothesis, H0: µ >= 8

And the alternative hypothesis is, H1: µ < 8

3) Assumtions and condition:

If the sample standard deviation is known, then we use T test.

If the population standard deviation is known, then we use Z test.

4) Test name: The alternative hypothesis is less than type.

Hence, the test is left tailed test.

Here, we are given the sample standard deviation.

Hence, we need to use 1 sample T test.

5) Test statistic:

Now, plug the values in the test statistic formula.

T = -3.62

That is, test statistic T = -3.62

Degrees of freedom = n-1 = 20 -1 = 19.

6) P-value:

Given: Significance level (α) = 0.01.

We need to find the p-value.

We can find out the p-value using technology or T table.

Suppose, we are using excel to find the p-value.
We can find out this by using the following excel command.

= T.DIST(Test statistic , Degrees of freedom, 1)

= T.DIST(-3.62 , 19, 1)

We will get, p-value = 0.000911

7) Make a decision:

The decision rule is,

If  P-value > α, then we fail to reject the null hypothesis.

If  P-value <= Test statistic, then we reject the null hypothesis.

p-value = 0.000911

α = 0.05

Here, P-value < α.

Hence, we reject the null hypothesis.

8) Conclusion about the claim:

Here, P-value < α.

Hence, we reject the null hypothesis.

Hence, we supports the claim statement.

That is, Mean waiting time is less than 8 minutes