Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a survey of a random sample of 35 households in the Cherry Creek neighborhood of Denver, it was found that 8 households turned out the lights and pretended not to be home on Halloween.

(a) Compute a 90% confidence interval for p, the proportion of all households in Cherry Creek that pretend not to be home on Halloween. (Round your answers to four decimal places.)

Lower limit:

Upper limit:

(b) What assumptions are necessary to calculate the confidence interval in part (a)?

np > 5 and nq < 5np < 5 and nq < 5     np < 5 and nq > 5np > 5 and nq > 5

(c) The national proportion of all households in the United States that turn out the lights and pretend not to be home on Halloween is 0.28. Is 0.28 in the confidence interval you computed?

Yes

No     

Based on your answer, does it seem that the Cherry Creek neighborhood is much different (either higher or lower proportion) from the population of all U.S. households? Explain.

It seems the Cherry Creek neighborhood has a higher proportion than the rest of the U.S. because 0.28 falls below the confidence interval.It seems the Cherry Creek neighborhood has a lower proportion than the rest of the U.S. because 0.28 falls above the confidence interval.     It seems the Cherry Creek neighborhood is not different from the rest of the U.S. because 0.28 is in the confidence interval.

Answer 1

Part a)

p̂ = X / n = 8/35 = 0.2286
p̂ ± Z(α/2) √( (p * q) / n)
0.2286 ± Z(0.1/2) √( (0.2286 * 0.7714) / 35)
Z(α/2) = Z(0.1/2) = 1.645
Lower Limit = 0.2286 - Z(0.1) √( (0.2286 * 0.7714) / 35) = 0.1118
upper Limit = 0.2286 + Z(0.1) √( (0.2286 * 0.7714) / 35) = 0.3453
90% Confidence interval is ( 0.1118 , 0.3453 )
( 0.1118 < P < 0.3453 )

Part b)

Condition
n * P >= 5   = 35 * 0.2286 = 8.001
n * (1 - P ) >= 5 = 35 * ( 1 - 0.2286 ) = 26.999

Both condition are satisfied

Part c)

90% Confidence interval is ( 0.1118 , 0.3453 )

Yes, value 0.28 lies in the interval, hence we fail to reject null hypothesis.

Part d)

  It seems the Cherry Creek neighborhood is not different from the rest of the U.S. because 0.28 is in the confidence interval.