Question

The number of hours worked per year per person in a state is normally distributed with a standard deviation of 39. A sample of 15 people is selected at random, and the number of hours worked per year per person is given below. Calculate the 98% confidence interval for the mean hours worked per year in this state. Round your answers to the nearest integer and use ascending order.Time205120612162216721692171218021832186219521962198220522102211

Answer 1

`2169.67-2.624(48.72)/(√(15))=2136.66`

`2169.67+2.624(48.72)/(√(15))=2202.68`

And the confidence interval would be given by (2137, 2203)

Explanation:

2051 ,2061 ,2162 ,2167 , 2169 ,2171 , 2180 , 2183 , 2186 , 2195 , 2196 , 2198 , 2205 , 2210 ,2211

We can calculate the mean and deviation with these formulas:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

`s=\sqrt{(\sum_(i=1)^n (x_i-\bar X))/(n-1)}` (3)

And we got:

`\bar X=2169.67` represent the sample mean for the sample

`\mu` population mean

s=48.72 represent the sample standard deviation

n=15 represent the sample size

Confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The degrees of freedom are given by:

`df=n-1=15-1=14`

Since the Confidence is 0.98 or 98%, the significance is
`\alpha=0.02` and
`\alpha/2 =0.01`, and using excel we calculate the critical value
`t_(\alpha/2)=2.624`

Now we have everything in order to replace into formula (1):

`2169.67-2.624(48.72)/(√(15))=2136.66`

`2169.67+2.624(48.72)/(√(15))=2202.68`

And the confidence interval would be given by (2137, 2203)