Question

Given a population in which the probability of success is p=0.55, if a sample of 400 items is taken, then complete parts a and b a. Calculate the probability the proportion of successes in the sample will be between 0.52 and 0.60. b. Calculate the probability the proportion of successes in the sample will be between 0.52 and 0.60 if the sample size is 200 a. The probability the proportion of successes in the sample will be between 0.52 and 0.60 is (Round to four decimal places as needed.) b. The probability the proportion of successes in the sample will be between 0.52 and 0.60 if the sample size is 200 is (Round to four decimal places as needed.)

Answer 1

a)

The standard deviation of the sample proportion is found as follows

6, - -2 - 0505) p(1-2) 0.55 (1 - 0.55) 400 h

σp  = 0.024874685

The probability the proportion of successes in the sample is 0.52 or less is found as

z = 0.52 -0.55 0.02487

0.52 - 0.55 z = -1.21 0.02487

Probability the proportion of successes in the sample is 0.52 or less equals 0.11314

The probability the proportion of successes in the sample is 0.60 or less is found as

0.60 - 0.55 Z 2.0 0.02487

Probability the proportion of successes in the sample is 0.60 or less .97725

Probability the proportion of successes in the sample will be between 0.52 and 0.60 = 0.97725 - 0.11314

Probability the proportion of successes in the sample will be between 0.52 and 0.60 is 0.8641

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b)

If the sample size is 200, the standard deviation of the sample proportion is found as

0,= |o«-2) – fa5(1. osa |p(1-p 0.55 (1 -0.55) 200 n

σp  = 0.035178118

The probability the proportion of successes in the sample is 0.52 or less is found as

| 0.52 -0.55 = 0.035178118 = -0.85

The probability the proportion of successes in the sample is 0.52 or less equals 0.19766

The probability the proportion of successes in the sample is 0.60 or less is found as

| 0.60 -0.55 = 0.035178118 = 1.42

The probability the proportion of successes in the sample is 0.60 or less equals 0.92220

Probability the proportion of successes in the sample will be between 0.52 and 0.60 = 0.92220 -   0.19766

Probability the proportion of successes in the sample will be between 0.52 and 0.60 if the sample size is 200 is 0.7245