Question

using the sample result ( \bar{x}=49.9 ) and ( s = 5.9 ) from a random sample of size ( n = 39 ), calculate the test statistic and p - value to test the following hypotheses and provide a conclusion at a significance level ( alpha=0.05 ). assume the data are normally distributed.
( h_{0}: mu=50 )
( h_{a}: mu<50 )
test statistic: - 0.11 (round your answer to two decimal places.)
p - value: (round your answer to four decimal places.)
what conclusion can you make at significance level ( alpha = 0.05 )?
a. do not reject ( h_{0} ). there is insufficient evidence to conclude that ( mu<50 ).
b. do not reject ( h_{0} ). there is sufficient evidence to conclude that ( mu<50 ).
c. reject ( h_{0} ). there is sufficient evidence to conclude that ( mu<50 ).
d. reject ( h_{0} ). there is insufficient evidence to conclude that ( mu<50 ).

Explanation:

Step1: Calculate the P - value

Since the test statistic \(t=- 0.11\) and the hypothesis is \(H_{a}:\mu<50\) (left - tailed test). For a \(t\) - distribution with \(n - 1=39 - 1 = 38\) degrees of freedom. Using a \(t\) - table or statistical software, the \(P\) - value for \(t=-0.11\) and \(df = 38\) is \(P(t<-0.11)\).
Since the \(t\) - value is close to \(0\), and for a \(t\) - distribution, \(P(t < - 0.11)\approx0.4562\) (using a \(t\) - table or calculator).

Step2: Make a conclusion

We compare the \(P\) - value with the significance level \(\alpha = 0.05\).
Since \(P - value=0.4562>0.05=\alpha\), we do not reject the null hypothesis \(H_{0}\).

Answer:

The \(P\) - value is approximately \(0.4562\). The conclusion is A. Do not reject \(H_{0}\). There is insufficient evidence to conclude that \(\mu<50\).