Question

The sugar content of the syrup in canned peaches is normally distributed. Suppose that the variance is thought to be σ2=18 (milligrams)2. A random sample of n=10 cans yields a sample standard deviation of s=4.8 milligrams. Part 1 (a) Test the hypothesis H0:σ2=18 versus H1:σ2≠18 using α=0.05 Find χ02 .

Answer 1

Answer:

$\chi^2 =\frac{10-1}{18} 23.04 =11.52$

The degrees of freedom are:

$df =n-1=10-1=9$

Now we can calculate the critical value taking in count the alternative hypotheis we have two values:

$\chi^2_{\alpha/2}= 2.70$

$\chi^2_{1-\alpha/2}= 19.02$

Since the calculated value is between the two critical values we FAIL to reject the null hypothesis and we can't conclude that the true variance is different from 18

Step-by-step explanation:

Information given

$n=10$ represent the sample size

$\alpha=0.05$ represent the confidence level  

$s^2 =4.8^2= 23.04$ represent the sample variance obtained

$\sigma^2_0 =18$ represent the value to verify

System of hypothesis

We want to verify if the true variance is different from 18, so the system of hypothesis would be:

Null Hypothesis: $\sigma^2 = 18$

Alternative hypothesis: $\sigma^2 \neq 18$

The statistic would be given by:

$\chi^2 =\frac{n-1}{\sigma^2_0} s^2$

And replacing we got:

$\chi^2 =\frac{10-1}{18} 23.04 =11.52$

The degrees of freedom are:

$df =n-1=10-1=9$

Now we can calculate the critical value taking in count the alternative hypotheis we have two values:

$\chi^2_{\alpha/2}= 2.70$

$\chi^2_{1-\alpha/2}= 19.02$

Since the calculated value is between the two critical values we FAIL to reject the null hypothesis and we can't conclude that the true variance is different from 18