Question

1) Consider two independent random samples of sizes n₁ = 14 and n₂ = 14, taken from two normally distributed populations. The sample standard deviations are calculated to be s₁= 1.98 and s₂ = 5.71, and the sample means are x¯1=-10.2and x¯2=-2.34, respectively. Using this information, test the null hypothesis H0:μ1=μ2against the one-sided alternative HA:μ1<μ2, using Welch's 2-sample t Procedure for independent samples.

a) Calculate the value for the t test statistic. Round your response to at least 2 decimal places.

b) The P-value is within which one of the following ranges? (Note the conservative estimate of the degrees of freedom (that is, DF is the lesser of n₁ - 1  and n₂ - 1), is used in the answer; if your calculator's range is incorrect, use tcdf(-E99,t,DF) to get the appropriate P-value).

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.01 < P-value < 0.05
D		0.005 < P-value < 0.01
E		P-value < 0.005

c) What is the most appropriate conclusion that can be made, at the 1% level of significance?

A		There is sufficient evidence to reject the null hypothesis in favour of the alternative, that the mean of Population 1 is less than that of Population 2.
B		We can be completely certain that the mean of Population 1 is less than the mean of Population 2, as the p-value is very small.
C		We can be completely certain the that means of the two populations are equal, as the p-value is very large.
D		There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the two population means are different.
E		The results of the hypothesis test are invalid, since the assumptions of the Welch approximate t procedure were not met.

2) Consider the following set of random measurements, taken from a normally distributed population before and after a treatment was applied.

Before Treatment	[7.38, 6.21, 7.9, 6.46, 7.56, 7.92]
After Treatment	[5.89, 6.54, 6.27, 6.52, 7.18, 6.98]
Difference	[1.49, -.33, 1.63, -.6E-1, .38, .94]

Test the null hypothesis H0:μD=0against the alternative hypothesis HA:μD≠0.

a) What is the value of the t test statistic? Remember to run a T test on just he difference list. Round your response to at least 3 decimal places.

b) What is the range in which the P-value falls?

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.025 < P-value < 0.05
D		0.010 < P-value < 0.025
E		P-value < 0.010

c) Is the null hypothesis rejected at:

a) the .05 level of significance? Yes or No

b) the .10 level of significance? Yes or No

Answer 1

The sample sizes are $n_1=14,n_2=14$ .

The hypotheses are

$H_0: \mu _1=\mu _2\\ H_1: \mu _1- \mu _2<0$

This is left-tailed T-test.

a)The test statistic is

$t=\frac{\overline{x_1}-\overline{x_2} }{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ t=\frac{-10.2+2.34 }{\sqrt{\frac{1.98^2}{14}+\frac{5.71^2}{14}}}\\ {\color{Blue} t=-4.87}$

b) The P-value is

$\textup{P-value}=P\left (T<-4.87 \right )\\ \textup{P-value}=0.000024$

Correct choice is (E) P-value < 0.005

c) The conclusion is

A

There is sufficient evidence to reject the null hypothesis in favor of the alternative, that the mean of Population 1 is less than that of Population 2.