Question

Frank, who lives in Texas, and his sister Lilly, who lives in Japan, correspond regularly. From what he can tell from the postmarks on both his and his sister's letters, it appears that it takes longer for Lilly's mal from Japan to reach him in Texas than it does for his letters from Texas to reach her in Japan. When Frank called his post office to ask if there was a reason for this, the postmaster told him that the delivery time of letters in both directions should be the same. Frank and his sister decided to collect data to see if letters from Japan to Texas take longer to be delivered than letters from Texas to Japan. They recorded the delivery time in days. After convincing themselves that the assumptions were reasonable, they performed a two-sample t-test and obtained the following computer output. Two sample T for To Texas vs To Japan Mean 8.74 6.75 StDev 2.92 SE Mean 0.84 0.85 To Tesas To Japan 95% CI for To Texas - mu To Japan: (-0.53.4.51) T-Test mu To Texas mu To Japun (s): T 1.66 P-0058 DF-18 Using a significance level of 0.05, which of the following statements best describes the conclusion that can be drawn from these data? There is convincing evidence that there is no difference in the mean delivery times. There is convincing evidence that there is a difference in the mean delivery times. There is convincing evidence that the mean delivery time from Japan to Texas is greater than the mean delivery time from Texas to Japan. There is not convincing evidence that the mean delivery time from Japan to Texas is greater than the mean delivery time from Texas to Japan. The t-test cannot be used for sample sizes that are this small.

Answer 1

Question:

The available options are:

(A) There is convincing evidence that there is no difference in the mean delivery times.

(B) There is convincing evidence that there is a difference in the mean delivery times.

(C) There is convincing evidence that the mean delivery time from Japan to Texas is greater than the mean delivery time from Texas to Japan.

(D) There is not convincing evidence that the mean delivery time from Japan to Texas is greater than the mean delivery time from Texas to Japan. (E) The t-test cannot be used for sample sizes that are this small.

Answer:

The correct option is;

(A) There is convincing evidence that there is no difference in the mean delivery times.

Step-by-step explanation:

Here we have the formula for the two-sample t-test given s;

$t =\frac{\overbar\overline{\rm x}_1 - \overbar\overline{\rm x}_2 }{\sqrt{\frac{\sigma ^2_1}{n_1} } +\frac{\sigma ^2_2}{n_2} }}$    

To Texas N₁ = 12,  $\overline{\rm x}_1$ =8.74 σ₁ = 2.92 SE Mean₁ = 0.84

To Japan N₂ = 9  $\overline{\rm x}_2$ = 6.75 σ₂ = 2.56 SE Mean₁ = 0.85

t =   1.667

Where the null hypothesis is

mu To Texas = mu To Japan and the alternative is

mu To Texas > mu To Japan

Here since the observed P value of P = 0.058 is greater than the significance level of 0.05 we fail to reject the null hypothesis and we will therefore, not accept the alternative hypothesis.

Hence, there is convincing evidence that there is no difference in the mean delivery times.