Question

The National Health Statistics Reports published in 2018 reported that a sample of 360 ten-year old boys had a mean weight of 70.5 pounds with a standard deviation of 5.3 pounds. In addition a sample of 329 ten-year-old girls had a mean weight of 68.7 pounds with a standard deviation of 4.3 pounds. Can you conclude that the mean weights of ten-year-old boys and girls differ? Use ???? = 0.01.

Answer 1

`t=\frac{70.5-68.7}{\sqrt{(5.3^2)/(360)+(4.3^2)/(329)}}}=4.913`

Since we conduct a bilateral test we have the p value given by:

`p_v =2*P(z>4.913)=8.97x10^(-7)`

Since the p value is very low compared to the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true means for girls and boys are different at 1% of significance

Explanation:

Information given

`\bar X_(boys)=70.5` represent the mean weigth for ten year boys

`\bar X_(girls)=68.7` represent the mean weigth for ten year girls

`s_(boys)=5.3` represent the sample deviation for 10 year boys

`s_(girls)=4.3` represent the sample standard deviation for 10 year girls

`n_(boys)=360` sample size for boys

`n_(girls)=329` sample size for girls

t would represent the statistic

`\alpha=0.01` significance level assumed

System of hypothesis to check

We need to conduct a hypothesis in order to check if the true means are different for boys and girls, the system of hypothesis would be:

Null hypothesis:
`\mu_(boys)=\mu_(girls)`

Alternative hypothesis:
`\mu_(boys) \\eq \mu_(girls)`

The statistic for this case would be given by:

`t=\frac{\bar X_(boys)-\bar X_(girls)}{\sqrt{(s^2_(boys))/(n_(boys))+(s^2_(girls))/(n_(girsl))}}` (1)

Replacing we got:

`t=\frac{70.5-68.7}{\sqrt{(5.3^2)/(360)+(4.3^2)/(329)}}}=4.913`

P value

We can assume that the degrees of freedom for this case are large enough to assume that the t distribution is approximately the normal distribution.

Since we conduct a bilateral test we have the p value given by:

`p_v =2*P(z>4.913)=8.97x10^(-7)`

Since the p value is very low compared to the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true means for girls and boys are different at 1% of significance