A statistics professor has asked his students to flip coins over the years. He h

Question

A statistics professor has asked his students to flip coins over the years. He has kept track of how many flips land heads and how many land tails. Combining the results of his students over many years, he has formed a 95% confidence interval for the long-run population proportion of heads to be (.497, .513).

a. Why is this interval so narrow?

b. Suppose he were to conduct a hypothesis test of whether the long-run population proportion of heads differs from one-half. Based on this interval (do not conduct the test), would he reject the null hypothesis at the .05 significance level? Explain briefly (no more than one sentence).

c. Does the interval provide strong evidence that the long-run population proportion of heads is much different from one-half? Explain briefly.

Explanation / Answer

a) the interval is narrow because in the long run, by the central limit theorem the probability of occurrence of head or tails equals. that is Probability of head or tail approaches to the population parameter values which is 0.5 for unbiased coin. thats why the interval is 0.497 < mean < 0.513

Margin of error = (upper limit - lower limit)/2
= (0.513 - 0.497)/2
= 0.008
Margin of error = z-score for 95% confidence*Standard error of proportion of heads
0.5 is the proportion of heads which results in maximum standard error of p
SEp = sqrt (p*(1-p)/n)
0.008 = 1.96*sqrt(0.5*0.5/sqrt n)
0.008 = 1.96*0.5 / sqrt n
sqrt n = 1.96*0.5/0.008 = 122.5
n = 122.5^2 = 15006.25
sample size = 15006

The confidence interval is very narrow because of the smaller Margin of error obtained on the basis of a very large sample.
In brief the reason for the narrower confidence interval is VERY LARGE SAMPLE of students enquired.

b) Null hypothesis will not be rejected because the confidence interval is narrow and contain the population parameter.

c) No,  interval doesn't provide strong evidence that the long-run population proportion of heads is much different from one-half.

Test conducted: [But not required]

Sample proportion = 0.513 - 0.008 = 0.505

Standard error of p = Margin of error/z-score for 95% confidence = 0.008/1.96 = 0.00408

z = (0.505-0.500)/0.00408 = 1.2254902

No, the Ho is not rejected because the calculated z value (1.22) < the z critical value (1.96)

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