Question

The weights, in pounds, of dogs in a city are normally distributed with a population standard deviation of 2 pounds and an unknown population mean. A random sample of 16 dogs is taken and results in a sample mean of 28 pounds. Identify the parameters needed to calculate a confidence interval at the 90% confidence level. Then find the confidence interval.

Answer 1

The confidence interval at the 90% confidence level for the population mean weight of dogs in the city is (27.1775, 28.8225) pounds.

How to find the confidence interval ?

To calculate a confidence interval at the 90% confidence level for the population mean weight of dogs in the city, you will need the following parameters:

Sample Size

Sample Mean

Population Standard Deviation

Confidence Level

Confidence Interval = x ± Z * (σ / √n)

Plug in the values from the question to be:

Confidence Interval = 28 ± 1.645 * (2 / 4)

Confidence Interval = 28 ± 0.8225

Lower bound:

= 28 - 0. 8225

= 27. 1775

Upper bound :

= 28 + 0. 8225

= 28. 8225

The confidence interval is therefore (27.1775, 28.8225) .

Answer 2

Parameters:

`z = 1.645, \sigma = 2, n = 16`

The 90% confidence interval for the weights, in pounds, of dogs in a city is between 27.1775 pounds and 27.8225 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 = (1-0.9)/(2) = 0.05`

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
`1-0.05 = 0.95`, so
`z = 1.645`

Now, find 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 = 1.645*(2)/(√(16)) = 0.8225`

The lower end of the interval is the sample mean subtracted by M. So it is 28 - 0.8225 = 27.1775 pounds

The upper end of the interval is the sample mean added to M. So it is 28 + 0.8225 = 27.8225 pounds

The 90% confidence interval for the weights, in pounds, of dogs in a city is between 27.1775 pounds and 27.8225 pounds.