Question

Stanley StatStudent is diligently working on a project for class. He has collected a random sample of data and calculates a 95% confidence interval for the proportion of all professional baseball players that are left-handed to be 25.42 % space plus-minus space 11.11 % . But now he is stuck. He doesn't know how to explain what this means.

1) Help Stanley out by writing a sentence to explain what the confidence interval he has calculated means in the context of the problem he is working on.

2) Stanley feels like his confidence interval is very wide. He wonders if it is possible to make the confidence interval more narrow. What would he have to do in order to calculate a more narrow confidence interval? (There are two ways to do this.)

3) Which of the two ways you suggested in #2 would be the better way for Stanley to narrow the confidence interval? Explain.

Answer 1

Solution:

     We are given that : Stanley StatStudent is diligently working on a project for class. He has collected a random sample of data and calculates a 95% confidence interval for the proportion of all professional baseball players that are left-handed to be 25.42 % space plus-minus space 11.11 % .

Part 1) Help Stanley out by writing a sentence to explain what the confidence interval he has calculated means in the context of the problem he is working on.

Find limits of confidence interval :

Lower limit = 25.42% - 11.11% = 14.31% and Upper limit = 25.42% + 11.11 = 36.53%

Thus it is 95% confidence that the true population proportion of all professional baseball players that are left-handed is in between 14.31% and 36.53%.

Part 2) Stanley feels like his confidence interval is very wide. He wonders if it is possible to make the confidence interval more narrow. What would he have to do in order to calculate a more narrow confidence interval? (There are two ways to do this.)

To reduce the length of confidence interval , we need to reduce Margin of Error , which is given by formula:

Thus Margin of Error E depends on Zc value which depends on confidence level c.

As well as E depends on sample size n.

Thus Either reduce the confidence level , so that Zc will reduce or increase the sample size n

So that Margin of error will decrease and hence length of confidence interval would be narrow.

Part 3) Which of the two ways you suggested in #2 would be the better way for Stanley to narrow the confidence interval? Explain.

We suggest to increase the sample size n , So that we can keep same confidence level. Reducing confidence level , will reduce the percent of confidence. That is making 90% instead of 95% , will be less confident.