Question

To test Ho: σ=4.6 versus H1: σ≠4.6​, a random sample of size n=11 is obtained from a population that is known to be normally distributed.

​(a) If the sample standard deviation is determined to be s=5.6​,

compute the test statistic.​

(b) If the researcher decides to test this hypothesis at the α=0.01 level of​ significance, use technology to determine the​ P-value.

​(c) Will the researcher reject the null​ hypothesis?

Answer 1

(a) The test-statistic for the given hypothesis for population standard deviation is as-

(a

(n 1)s2 -2 2
, and degrees of freedom,
df n-1
.

Given:

sample standard deviation, s 5.6

sample size, n 11

g 4.6

(n 1)s (11 1)5.62 14.82041588 2 4.62

So, the test-statistics is computed as

14.82041588

(b) The level of significance is given as
= 0.0
, so we need to find the p-value for the calculated test-statistic and then we have to compare it with the significance level $\alpha$ to make decision.

14.82041588, df n- 1- 11 1 10

2P2 14.82041588, df 10) 2(0.1387476) 0.2774953 p -value _

So, the p-value is calculated as

p value 0.2774953

(c)
0.01, p-value 0.2774953

Since, $\mathrm{p-value>\alpha}\Rightarrow \mathbf{We\:fail\:to\:reject\:null\:hypothesis\:H_{0}}$

So, at
= 0.0
the sample does not provide enough evidence to support the alternative hypothesis, hence we fail to reject the null hypothesis, $\mathrm{H_{0}:\sigma=4.6}$