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?

Consider the following calculations

Explanation:

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
=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
=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.