Question

The population mean of the heights of five-year old boys is 100 cm. A teacher measures the height of her twenty-five students, obtaining a mean height of 105 cm and standard deviation 18. Perform a test with a 5% significance level to calculate whether the true mean is actually greater than 100cm

Answer 1

Here we are required to test whether the true mean or population mean µ is greater than 100 cm.

So the null hypothesis H0: µ =100 vs H1: µ >100 ( The claim)

The sample mean = X̄ =105 cm

sample standard deviation = s=18

sample size = n= 25

So, here the population standard deviation is unknown.

Assuming normality of the data, a one-sample t test needs to performed.

The test statistic is:

t= √n * (X̄ -µ0)/ s follows a  t distribution with (n-1) degrees of freedom.

where µ0 is the hypothesized mean =100

So Test statistic t= √n * (X̄ -µ0)/ s =√25 *(105-100) / 18 = 1.3889

Critical value and p-value:

The critical value for t at alpha=0.05 for degrees of freedom =n-1=25-1=24 is

t(0.05,24)=1.711

For the one-taile test, the p value for the observed t=1.3889 for degrees of freedom =24 is

P(t>1.3889)=0.088

Decision:

The test statistic t=1.3889 is < t critical value=1.711

and p-value=0.088 is > alpha=0.05

Hence, we fail to reject the null hypothesis at level of significance 0.05.

Conclusion:

We don't have enough evidence to conclude that the true mean is greater than 100 cm.