Question

a) A researcher claims that the mean age of CEOs (chief executive officers) of all corporations in the United States is 4646 years. A sample of 5050 corporations showed that the mean age of their CEOs is 48.348.3 years with a standard deviation of 5.55.5 years. Find the pp-value for the test that the mean age of CEOs of all corporations is different from 4646 years at αα = 0.0020.002.

b) The manager of a bank claims that the mean waiting time for all customers at that bank is not more than 1212 minutes. A sample of 2020 customers who visited this bank gave a mean waiting time of 14.7514.75 minutes and a standard deviation of 3.43.4 minutes. Test at the 2.5%2.5% significance level if the mean waiting time for all customers who visit this bank is greater than 1212 minutes. State the assumption(s) required for the test.

Answer 1

## a ) A researcher claims that the mean age of CEOs (chief executive officers) of all corporations in the United States is 46 years. A sample of 50 corporations showed that the mean age of their CEOs is 48.3 years with a standard deviation of 5.5 years. Find the pp-value for the test that the mean age of CEOs of all corporations is different from 46 years at αα = 0.002.

Answer :

we have given :

n  = sample size  = 50

x̄ = sample mean  = 48.3

μ = population mean = 46

s = sample  standard deviation  = 5.5

α = 0.2 % = 0.002

Claim  : Test to determine if the mean age of CEOs of all corporations is different from 46 years

## step1 : null and alternative Hypothesis :

Ho : μ = 46 vs H1: μ ≠ 46
( it is two tailed test )

## step 2 : Test statistics :

t = (x̄ - μ) * sqrt ( n ) / s

t = ( 48.3  - 46 ) * sqrt (50) / 5.5

= 16.2634 / 5.5

= 2.9569

## step 3 :  level of significance  = 0.002

## step 4 :  P value : it is two tailed test :

df = degree of freedom = n -1 = 49

= 2 * P ( t > 2.9569)   ( use statistical table )

= 2 * 0.002391  

= 0.004782

step 5 : Decision  :

We reject Ho if p value is less than alpha value using p value approach here
p value is greater  than alpha value we fail to reject Ho at given level of significance .

Step 6 : Conclusion :

There is Insufficient evidence to conclude that   the mean age of CEOs of all corporations

is different from 46 years .

that is we accept Ho , and mean is equal to 46 .

### b) The manager of a bank claims that the mean waiting time for all customers at that bank is not more than 1212 minutes. A sample of 2020 customers who visited this bank gave a mean waiting time of 14.7514.75 minutes and a standard deviation of 3.43.4 minutes. Test at the 2.5%2.5% significance level if the mean waiting time for all customers who visit this bank is greater than 1212 minutes. State the assumption(s) required for the test.

Answer : we have given :

n  = sample size  = 20

x̄ = sample mean  = 14.75

μ = population mean = 12

s = sample  standard deviation  = 3.4

α = 2.5 % = 0.025

Claim  : Test to determine if the mean waiting time for all customers who visit this bank is greater than 12 minutes .

# it is one sample t test : here sample size < 30 and population variance unknown here all the assumptions or condition satisfy for t test hence it used .

## step1 : null and alternative Hypothesis :

Ho : μ = 12 vs H1: μ > 12

( it is one tailed test , right tailed test )

## step 2 : Test statistics :

t = (x̄ - μ) * sqrt ( n ) / s

t = ( 14.75 - 12 ) * sqrt (20) / 3.4

= 12.2983   / 3.4

= 3.6171

## step 3 :  level of significance  = 0.025

## step 4 :  P value : it is one  tailed test :

df = degree of freedom = n -1 = 19

= P ( t > 3.6171)   ( use statistical table )

= 0.00091

step 5 : Decision  :

We reject Ho if p value is less than alpha value using p value approach here
p value is less than alpha value we reject Ho at given level of significance .

Step 6 : Conclusion :

There is sufficient evidence to conclude that   the mean waiting time for all customers who visit this bank is greater than 12 minutes .