A 95% confidence intervals for the proportion of first-year...

A 95% confidence intervals for the proportion of first-year statistics students with brown hair is given to be (0.25, 0.42). (a) What is the sample proportion of students with brown hair? (Round to 3 decimal places, if needed.) (b) What is the margin of error of this interval? (Round to 3 decimal places, if needed.) (c) What is the critical value used for this interval? (Round to 3 decimal places, if needed.) (d) What is the standard error of the estimate? (Round to 3 decimal places, if needed.) (e) What was the sample size? (Round to the appropriate integer.)

Transcript

00:01 In this question, we have been given that the value of confidence interval is here equals to it is 0 .25.

00:07 Now it will be here comma 0 .42 and here in the first part of this very problem we can say that confidence interval is equals to it is p hat plus minus it is z critical multiplied with under root of p hat and then it is 1 minus p hat.

00:24 Now in the denominator it is sample size.

00:26 So we will be getting the value over here as so it will be 0 .25 and then it is comma 0 .42.

00:33 So if we have to calculate the value of p hat what we can do is we can simply add both of these lower limit and upper limit.

00:41 So it will be 0 .25 and then it is here plus 0 .42 divided by 2 and it will be here equals to 0 point and then it is 3.

00:51 Now it will be here 3 now it is 5.

00:53 So we can put this very value inside the box and then we will proceed towards the next part of this very problem and here the value of z critical for 95 percentage confidence level is 1 .96 and the value of here alpha is definitely 1 minus 0 .95 and that is 0 .05.

01:12 Now here it can be said that let us write it will be the value of 0 point and then it is 335 minus it will be here 1 .96.

01:23 So we are equating the value of lower limit and here it will be 0 .335 multiplied with 1 minus 0 .335 and then in the denominator it is n value.

01:34 So that is here equals to 0 .25 and then we can say that 0 .25 and now it will be here what it is going to be negative and then it is plus 0 .335 and here it will be equals to let us write under root of it is going to be 0 .335 and then it is multiplied with 1 minus and then it will be 0 .335 and then in the denominator we will be having the value of sample size.

02:04 So what will happen is we can here say that the value of sample size can be here obtained by writing it as 0 .335 and then it is minus 0 .25.

02:16 Now it will be here raised to the power 2 and that is here 0 .335 multiplied with 1 minus 0 .335 and then in the denominator it is n...

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