Question

The heights of dogs, in inches, in a city are normally distributed with a population standard deviation of 7 inches and an unknown population mean. If a random sample of 20 dogs is taken and results in a sample mean of 21 inches, find a 95% confidence interval for the population mean. z0.10z0.10 z0.05z0.05 z0.025z0.025 z0.01z0.01 z0.005z0.005 1.282 1.645 1.960 2.326 2.576 You may use a calculator or the common z values above. Round the final answer to two decimal places.

Answer 1

The 95% confidence interval is
`17.932  <  \mu <  20  + 22.068`

Explanation:

From the question we are told that

The population standard deviation is
`\sigma =7 \ inches`

The sample size is n = 20

The sample mean is
`\= x = 20`

From the question we are told the confidence level is 95% , hence the level of significance is

`\alpha = (100 - 95 ) \%`

=>
`\alpha = 0.05`

Generally from the normal distribution table the critical value of
`(\alpha )/(2)` is

`Z_{(\alpha )/(2) } =  1.96`

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } *  (\sigma )/(√(n) )`

=>
`E = 1.96 *  (7 )/(√(20) )`

=>
`E = 2.068`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  \mu <  \=x  +E`

=>
`20  - 2.068  <  \mu <  20  + 2.068`

=>
`17.932  <  \mu <  20  + 22.068`