A 90% confidence intervals for the proportion…

Question

A 90% confidence intervals for the proportion of first-year statistics students with brown hair is given to be (0.25, 0.41).

(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)

Answer 1

I am going to start at the beginning, which almost always is definitions.

Let us start with a proportion

proportion: a statistical proportion is a ratio, typically the ratio of a given trait or number of successes to the total population. Example. p = s / n where s = successes and n = number

confidence interval (CI) is a range of estimates for an unknown parameter, defined as an interval with a lower bound and an upper bound. For symmetric distributions like proportions, the mean is half way between the lower and upper bounds

margin of error (MOE): for this problem the confidence interval is defined as the mean value ± the MOE. Thus the span or width of the CI is equal to twice the MOE.

With these definitions we can answer the first two questions.

a) the expected value of the proportion will be in the middle of the given CI of [.24 .41]. This is .325

b) the margin of error is the difference between either the lower limit and the mean or the mean and the upper limit. The answer is the same (.325-.24) or ( .41 - .325) which is. .085 (MOE)

This is the HOW for your question. First find the definitions that are needed then use them to find the bits and pieces of the answer.

I'll give some more formulae or definitions that you will need for the rest of the questions.

critical value (z_crit): a value dependent on the statistical distribution which will meet specified requirements of confidence or error. For proportions we use z distribution. You can look up the two sided test critical value of Z for an alpha of .10 online or use a calculator. hint: use (1-α/2) I looked it up and Z_.95 = 1.644

standard deviation of a proportion (s): formula for s is defined as s = √[p(1-p)]. where p is expected value of proportion. (calculated in a)

standard error: This is the same as standard deviation of the mean and has the following formula

se = s / √n where n = number of samples

CI formulae: using MOE we can define CI as p ± MOE.

remember: MOE is also defined as z_{crit *}se.

Using these formulae, definitions, and some algebra, you can calculate the sd (s) from the given data. I gave you the Z crit and from this you can calculate the se. the n can be calculated using algebra from the se formula that I gave.

This is like a jigsaw puzzle where you have to work out the parts one at a time. You have the tools now. They should have been in your text book or class notes if you had been taking good notes.

Answer 2

Hi Mish L.,

You've asked 7 questions, in great detail, on the same skill set, all at the same time, without evidence you've even thought about the problems? That's spamming. Doing your entire problem set for you is first, forbidden under Wyzant rules, and second, won't help you acquire the skills you need. Ask one question, digest the lesson, and do the rest yourself -- that would be appropriate.

-- Cheers, --Mr. d.