9.5.111 t-test

A sample​ mean, sample​ size, and sample standard deviation are provided below. Use the​ one-mean t-test to perform the required hypothesis test at the 1​% significance level.

\(\bar x\) = 21, s = 10, n = 32, $ \(H_0: \mu =27, H_a: \mu \neq 27\)

The test statistic is t =

First, we need to get the data the question

x = 21
s = 10
n = 32
mu = 27

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To get the test statistic t we use the formular \(t = \frac{\bar x-\mu}{\frac{s}{\sqrt{n}}}\), we run

t = (x-mu)/(s/sqrt(n))
t

## [1] -3.394113

Round the answer to two decimal places

round(t,2)

## [1] -3.39

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Identify the critical​ value(s). Select the correct choice below and fill in the answer box within your choice.(Round to three decimal places as​ needed.).

First, this is a two tailed test since \(H_a: \mu \neq 27\), we have two critical values.

We need to find degree of freedom

deg = n-1

Since the significance level is 1% and this is a two tail test, the area to the left of a negative critical will be .01/2

We can find the negative critical value by use qt() command

qt(.01/2, deg)

## [1] -2.744042

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Round to three decimal places

round(qt(.01/2, deg),3)

## [1] -2.744

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Since our test statistic t = -3.39 < our critical value \(t_{\alpha/2}\) = -2.744, out test statistic lies in the rejected region. So we have enough evidence to reject null hypothesis.

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Hope that helps!