People end up tossing 12% of what they buy at the grocery store (Reader's Digest, March, 2009). Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.a- Show the sampling distribution of ( p¯ ), the proportion of groceries thrown out by your sample respondentsb- what is the probability that the sample proportion will be within ±.02 of the population proportion?c- what is the probability that your survey will provide a sample proportion within ±.015 of the population proportion?

People end up tossing 12% of what they buy at the grocery store (Reader's Digest, March, 2009). Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.

a- Show the sampling distribution of ( p¯ ), the proportion of groceries thrown out by your sample respondents

sampling distribution of ( p¯ ) is normal with

mean = 0.12 and

standard error = sqrt(p(1-p)/n) = sqrt(0.12*0.88/540) =0.0140

b- what is the probability that the sample proportion will be within ±.03 of the population proportion?

z value for 0.03 difference, z=0.03/0.014 =2.14

The required P= P( -2.14<z<2.14) = P( z <2.14) – P( z <-2.14)

=0.9838 - 0.0162

=0.9676

c- what is the probability that your survey will provide a sample proportion within ±.015 of the population proportion?

z value for 0.015 difference, z=0.015/0.014 =1.07

The required P= P( -1.07<z<1.07) = P( z <1.07) – P( z <-1.07)

=0.8577 - 0.1423

=0.7154

d- What would be the effect of taking a larger sample on the probabilities in parts (b) and (c)? Why?

Taking a larger sample will decrease the standard error. The probabilities in parts (b) and (c) will increase.

abiolataiwo2015 abiolataiwo2015 · Answer 2 · 2022-03-29T15:36:12+02:00

The sampling distribution of ( p¯ ), the proportion of groceries thrown out by your sample respondents is: .01398 .

Sampling distribution

Probability of occurrence=p = .12

Probability of not getting the occurrence=q = 1 - .12 = .88

Sample size=n = 540

a. Standard error of the distribution of sample means

s =√(p×q/n)

s=√(.12×.88/540)

s= .01398

b. z=0.03/0.014 =2.14

P= P( -2.14<z<2.14) = P( z <2.14) – P( z <-2.14)

P=0.9838 - 0.0162

P=0.9676

c. z1 = -.015/.01398

z1 = -1.07 = the low z-score

z2 = +.015/.01398

z2 = +1.07 = the high z-score

p(z-score is between -1.07 and 1.07) = .7154

Using the z-score table

z1 area under the distribution curve = .1423

z2 area under the distribution curve = .8577

Hence:

z2 - z2 area = .8577 - .1423

z2 - z2 area = .7154

d- The effect of taking a larger sample on the probabilities in parts (b) and (c) is: Taking a larger sample will reduce the standard error and the probabilities in parts (b) and (c) will increase.

Inconclusion the sampling distribution of ( p¯ ), the proportion of groceries thrown out by your sample respondents is: .01398 .

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