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

Answer:

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

Step-by-step explanation:

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

$\alpha = \frac{1 - 0.9}{2} = 0.05$

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

$M = z\frac{\sigma}{\sqrt{n}}$

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

$M = 1.645\frac{23.44}{\sqrt{141}}$

$M = 3.25$

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).