Question

According to an estimate, 2 years ago the average age of all CEOs of medium-sized companies in the United States was 58 years. Jennifer wants to check if this is still true. She took a random sample of 70 such CEOs and found their mean age to be 54 years with a standard deviation of 6 years. Using the 1% significance level, can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years?

Answer 1

Explanation:

We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 58

For the alternative hypothesis,

µ ≠ 58

This is a two tailed test.

Since the population standard deviation is not given, the distribution is a student's t.

Since n = 70

Degrees of freedom, df = n - 1 = 70 - 1 = 69

t = (x - µ)/(s/√n)

Where

x = sample mean = 54

µ = population mean = 58

s = samples standard deviation = 6

t = (54 - 58)/(6/√70) = - 5.58

We would determine the p value using the t test calculator. It becomes

p < 0.0000

Since alpha, 0.01 > than the p value, then we would reject the null hypothesis. Therefore, at a 1% significance level, we can conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years.

Answer 2

`t=(54-58)/((6)/(√(70)))=-5.58`

Now we can calculate the degrees of freedom:

`df =n-1= 70-1=69`

And the p value would be given by this probability taking in count the bilateral test:

`p_v =2*P(t_(69)<-5.58)=4.38x10^(-7)`

Since the p value is lower than the significance level provided we have enough evidence to reject the null hypothesis and we can conclude that the true mean is different from 58

Explanation:

Information given

`\bar X=54` represent the mean age for the CEOs

`s=6` represent the sample deviation

`n=70` sample size

`\mu_o =58` represent the value to verify

`\alpha=0.01` represent the significance level

t would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to verify if mean age of all CEOs of medium-sized companies in the United States is different from 58 years, the system of hypothesis would be:

Null hypothesis:
`\mu = 58`

Alternative hypothesis:
`\mu \\eq 58`

The statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

Replacing the info given we got:

`t=(54-58)/((6)/(√(70)))=-5.58`

Now we can calculate the degrees of freedom:

`df =n-1= 70-1=69`

And the p value would be given by this probability taking in count the bilateral test:

`p_v =2*P(t_(69)<-5.58)=4.38x10^(-7)`

Since the p value is lower than the significance level provided we have enough evidence to reject the null hypothesis and we can conclude that the true mean is different from 58