Question

A survey was conducted that asked 997 people how many books they had read in the past year. Results indicated that x overbar equals 13.2 books and sequels 18.9 books. Construct a 95 ​% confidence interval for the mean number of books people read. Interpret the interval.

Answer 1

The 95% confidence interval would be given by (12.03;14.37)

Explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X=13.2` represent the sample mean

`\mu` population mean (variable of interest)

s=18.9 represent the sample standard deviation

n=997 represent the sample size

2) Calculate the confidence interval

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

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

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=997-1=996`

Since the confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,996)".And we see that
`t_(\alpha/2)=1.96` and this value is exactly the same for the normal standard distribution and makes sense since the sample size is large enough to approximate the t distribution with the normal standard distribution.

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

`13.2-1.96(18.9)/(√(997))=12.03`

`13.2+1.96(18.9)/(√(997))=14.37`

So on this case the 95% confidence interval would be given by (12.03;14.37)