Question

The amount of time required to reach a customer service representative has a huge impact on customer satisfaction. Below is the Excel output from a study to see whether there is evidence of a difference in the mean amounts of time required to reach a customer service representative between two hotels. Assume that the population variances in the amount of time for the two hotels are not equal.
t-Test: Two-sample assuming Unequal Variances

Hotel 1 Hotel 2
Mean 2.56 2.01
Variance 2.952 3.579
Observations 20 20
df 38
t Stat ?
t Critical one-tail 1.686
t Critical two-tail 2.024

Required:
What is the value of the test statistic?

Answer 1

Answer:

The value of the test statistic is t=1.12.

Step-by-step explanation:

This is a hypothesis test for the difference between populations means.

The claim is that the mean amount of time required to reach a customer service representative significantly differs between the two hotels.

Then, the null and alternative hypothesis are:

$H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0$

The sample 1, of size n1=20 has a mean of 2.65 and a standard deviation of √2.952=1.72.

The sample 2, of size n2=20 has a mean of 2.01 and a standard deviation of √2.952=1.89.

The difference between sample means is Md=0.64.

$M_d=M_1-M_2=2.65-2.01=0.64$

The estimated standard error of the difference between means is computed using the formula:

$s_{M_d}=\sqrt{\dfrac{\sigma_1^2+\sigma_2^2}{n}}=\sqrt{\dfrac{1.72^2+1.89^2}{20}}\\\\\\s_{M_d}=\sqrt{\dfrac{6.531}{20}}=\sqrt{0.327}=0.571$

Then, we can calculate the t-statistic as:

$t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{0.64-0}{0.571}=\dfrac{0.64}{0.571}=1.12$