10.2.39 pooled t-test two-tailed

Provided below are summary statistics for independent simple random samples from two populations. Use the pooled​ t-test and the pooled​ t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.

\(\bar x_{1} = 15, s_1=2.3, n_1=15\)

\(\bar x_{2} = 16, s_2=2.2, n_2=15\)

(a).​ First, what are the correct hypotheses for a​ two-tailed test?

Since it is a two tailed test, the correct hypothese is

\(H_0: \mu_1 =\mu_2\)

\(H_a: \mu_1 \neq \mu_2\)

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Next, compute the test statistic. (Round to three decimal places as​ needed.)

First, we need to get the data from the question

x1 = 15
s1 = 2.3
n1 = 15
x2 = 16
s2 = 2.2
n2 = 15

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This is a pooled t-test, to compute pooled sample standard deviation we use the formula \(s_p= \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}\)

sp = sqrt( ( (n1-1)*s1^2 + (n2-1)*s2^2 )/ (n1+n2-2) )
sp

## [1] 2.250555

To get the test statistic t we use the formular \(t= \frac{\bar x_1-\bar x_2}{s_p \sqrt{1/n_1+1/n_2}}\), we run

t = (x1-x2)/(sp*sqrt(1/n1+1/n2))
t

## [1] -1.216861

Round the answer to three decimal places

round(t,3)

## [1] -1.217

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Now determine the critical values. (Round to three decimal places as​ needed.)

We need to find degree of freedom

deg = n1+n2-2

Since this is a two-tailed test and \(\alpha = .01\), the area to the left of a negative critical value is equal to \(\alpha/2\)

alpha = .01
t_critical = abs(qt(alpha/2,deg))
t_critical

## [1] 2.763262

Round the answer to three decimal places

round(t_critical,3)

## [1] 2.763

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Since our test statistic t = -1.217 > our negative critical value \(t_{\alpha/2}\) = -2.763 and < our positive critical value \(t_{\alpha/2}\) = 2.763, out test statistic does not lie in the rejected region. So we do not have enough evidence to reject null hypothesis.

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(b) The 99% confidence interval is from… to…​(Round to three decimal places as​ needed.)

99% confidence interval means \(\alpha = .01\) so \(t_{\alpha/2}=t_{critical value}\)

To find confidence interval we use the formular \((\bar x_1 - \bar x_2) \pm t_{\alpha/2}.s_p\sqrt{1/n_1+1/n_2}\)

(x1-x2) - t_critical*sp*sqrt(1/n1+1/n2)

## [1] -3.270812

(x1-x2) + t_critical*sp*sqrt(1/n1+1/n2)

## [1] 1.270812

Round to three decimal places

round((x1-x2) - t_critical*sp*sqrt(1/n1+1/n2),3)

## [1] -3.271

round((x1-x2) + t_critical*sp*sqrt(1/n1+1/n2),3)

## [1] 1.271

We finish a lot of complicated work by using R

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Hope that helps!