Question

{Exercise 9.09 (Algorithmic).} Consider the following hypothesis test: Ho: u 20 Ha: p < 20 A sample of 40 provided a sample mean of 19.6. The population standard deviation is 1.8. a. Compute the value of the test statistic (to 2 decimals). -1.64 b. What is the p-value (to 3 decimals)? -1.757 X c. Using a = .05, can it be concluded that the population mean is less than 20? No d. Using a = .05, what is the critical value for the test statistic? e. State the rejection rule: Reject Ho if z is less than or equal to the critical value. f. Using a = .05, can it be concluded that the population mean is less than 20? No

Answer 1

Solution :

$\mu$= 20

$\bar x$ = 19.6

$\sigma$ = 1.8

n = 40

This is the left tailed test .

The null and alternative hypothesis is ,

H₀ :  $\mu$ ≥ 20

H_a : $\mu$ < 20

a ) Test statistic = z

= ($\bar x$ - $\mu$ ) / $\sigma$ / $\sqrt$ n

= (19.6 - 20 ) / 1.8 / $\sqrt$ 40

= −1.405

Test statistic = z =  −1.40

b ) P-value =0.0799

c ) the population mean μ is ;less than 20, at the 0.05 significance level

d )The significance level is α = 0.05, and the critical value for a left tailed test is z_c ​= −1.64

e ) It is observed that z = -1.405 ≥ zc​=−1.64, it is then concluded that the null hypothesis is not rejected.

$\alpha$ = 0.05  

P-value ≥ $\alpha$

0.0228 ≥ 0.05

f ) The null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is less than 20, at the 0.05 significance level