To test a hypothesis, it means we want to verify or refute a claim. Here, we want to verify or refute whether the new method is better or not.
The new method is better, and the company should not take any action.
The given parameters are:
[tex]\alpha = 0.05[/tex] -- significance level
[tex]\sigma =32.2[/tex] -- the sample standard deviation
The claim that the standard deviation is greater than 32.2 means that:
The alternate hypothesis is:
[tex]H_a: \sigma > 32.2[/tex]
The null hypothesis is:
[tex]H_o: \sigma = 32.2[/tex]
So, we have:
[tex]H_o: \sigma = 32.2[/tex] --- null hypothesis
[tex]H_a: \sigma > 32.2[/tex] --- alternate hypothesis
Calculate the mean value
[tex]\bar x = \frac{\sum x}{n}[/tex]
[tex]\bar x = \frac{-41+79-22-71-42+13+18+53-6-50-107-107}{12}[/tex]
[tex]\bar x = \frac{-283}{12}[/tex]
[tex]\bar x = -23.583[/tex]
Calculate the sample standard deviation
[tex]\sigma_x =\sqrt{\frac{\sum(x - \bar x)^2}{n-1}}[/tex]
This gives:
[tex]\sigma_x =\sqrt{\frac{(-41--23.583)^2+(79--23.583)^2+.......+(-107--23.583)^2}{12-1}}[/tex]
[tex]\sigma_x =\sqrt{\frac{37272.916668}{11}}[/tex]
[tex]\sigma_x =\sqrt{3388.44696982}[/tex]
[tex]\sigma_x =58.210368233[/tex]
Approximate
[tex]\sigma_x =58.2[/tex] --- to 1 decimal place
Next, calculate the test statistic using:
[tex]t = \frac{(n - 1) * \sigma_x^2}{\sigma^2}[/tex]
[tex]t = \frac{(12 - 1) * 58.2^2}{32.2^2}[/tex]
[tex]t = \frac{11 * 58.2^2}{32.2^2}[/tex]
[tex]t = 35.94[/tex]
Calculate the degrees of freedom (df)
[tex]df=n-1[/tex]
[tex]df=12-1[/tex]
[tex]df=11[/tex]
So, we calculate the p-value with the following parameters
[tex]df=11[/tex]
[tex]t = 35.94[/tex]
The p value is
[tex]p \approx 0[/tex]
Because the p value is less than the significance level
i.e. [tex]0 < 0.05[/tex]
We have to reject the null hypothesis and accept the alternative hypothesis
This means that the new method is better, and the company should not take any action.
Learn more about test of hypothesis at:
https://brainly.com/question/10758924
