Question

A research company desires to know the mean consumption of meat per week among people over age 29. A sample of 2092 people over age 29 was drawn and the mean meet consumption was 2.9 pounds. Assume that the standard deviation is known to be 1.4 pounds. Construct a 95% confidence interval for the mean consumption of meat among people over age 29. Round your answer to one decimal place.

Answer 1

`2.9-1.96(1.4)/(√(2092))=2.84`

`2.9+1.96(1.4)/(√(2092))=2.96`

The confidence interval is given by
`2.84 \leq \mu \leq 2.96`

Explanation:

Information given

`\bar X=2.9` represent the sample mean

`\mu` population mean

`\sigma =1.4` represent the population standard deviation

n=2092 represent the sample size

Confidence interval

The confidence interval is given by:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

Since the Confidence interval is 0.95 or 95%, the significance is
`\alpha=0.05` and
`\alpha/2 =0.025`, and the critical value would be
`z_(\alpha/2)=1.96`

Replacing the info we got:

`2.9-1.96(1.4)/(√(2092))=2.84`

`2.9+1.96(1.4)/(√(2092))=2.96`

The confidence interval is given by
`2.84 \leq \mu \leq 2.96`

Answer 2

The 95% confidence interval for the mean consumption of meat among people over age 29 is between 2.8 pounds and 3 pounds.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.95)/(2) = 0.025`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.025 = 0.975`, so
`z = 1.96`

Now, find the margin of error M as such

`M = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

`M = 1.96*(1.4)/(√(2092)) = 0.1`

The lower end of the interval is the sample mean subtracted by M. So it is 2.9 - 0.1 = 2.8 pounds

The upper end of the interval is the sample mean added to M. So it is 2.9 + 0.1 = 3 pounds.

The 95% confidence interval for the mean consumption of meat among people over age 29 is between 2.8 pounds and 3 pounds.