Question

Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 141 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.09 with a standard deviation of 23.44

Construct a 90% confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places A 90% confidence interval for the mean number of letter sounds identified in one minute is:______

Answer 1

A 90% confidence interval for the mean number of letter sounds identified in one minute is: (30.84, 37.34).

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`.

That is z with a pvalue of
`1 - 0.05 = 0.95`, so Z = 1.645.

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 = 1.645(23.44)/(√(141))`

`M = 3.25`

The lower end of the interval is the sample mean subtracted by M. So it is 34.09 - 3.25 = 30.84.

The upper end of the interval is the sample mean added to M. So it is 34.09 + 3.25 = 37.34.

A 90% confidence interval for the mean number of letter sounds identified in one minute is: (30.84, 37.34).