Question

The waiting time for patients at local walk-in health clinic follows a normal distribution
with a mean of 15 minutes and a population standard deviation of 5 minutes. The
quality-assurance department found in a sample of 64 patients that the mean waiting
time was 13.5 minutes. Using the 99% confidence level and the 95% confidence interval,
is it reasonable to conclude the sample mean waiting time is statistically significantly
different from the population mean waiting time?
i. Firstly, what are the null and the alternative hypothesis?
ii. What are the critical values at 1% and 5% significance level (two-tailed test)?
iii. What is the value of the test statistic (z obtained)?
iv. Finally, state your decision in terms of the null hypothesis.

Answer 1

Answer :

We have given :  

n = sample size = 64

x̄ = sample mean  = 13.5 Minutes

μ = population mean = 15 Minutes

σ = population standard deviation  = 5 Minutes

α =  99% confidence level and the 95% confidence interval

Claim  :

is it reasonable to conclude the sample mean waiting time is statistically significantly
different from the population mean waiting time?

## i. Firstly, what are the null and the alternative hypothesis?

Answer :

Ho : μ = 15 VS H1 : μ ≠ 15 ( claim)

it is two tailed test

## ii. What are the critical values at 1% and 5% significance level (two-tailed test)?

Answer : for 99 % level critical value = - 2 .58 to 2.58 and

for 95 % level critical value = - 1.96 to 1.96

## iii. What is the value of the test statistic (z obtained)?

Answer : z = ( (x̄ - μ) *√ n / σ

z = ( 13.5  - 15 ) * √ 64   / 5

z = - 12 / 5

z = - 2.4

## Decision :

We reject Ho at ( 1 - α ) * 100 % level of significance if z stat < - z critical value or

z stat > z critical value ,

for α =  99% confidence level : z stat > - z critical value hence we fail to reject Ho ie - 2.4 > - 2.58 and

for α =   95% confidence interval : z stat <  - z critical value   hence we reject Ho ,   - 2 .4 < - 1.96 .

## iv. Finally, state your decision in terms of the null hypothesis.

Answer :

## for α =  99% confidence level  

conclusion :

There is Insufficient evidence to conclude that   the sample mean waiting time is statistically significantly different from the population mean waiting time . here we Accept Ho

## for α =  95 % confidence level  

conclusion :

There is sufficient evidence to conclude that   the sample mean waiting time is statistically significantly different from the population mean waiting time . here we Reject   Ho