Sixty percent of all grocery shoppers at the local Trader Joes purchase at least

Question

Sixty percent of all grocery shoppers at the local Trader Joes purchase at least one organic product as part of their shopping order. A random sample of 400 shoppers at this Trader Joes is taken.
1What is the sampling distribution of the sample proportion? On what do you base your answer?
2What is the probability that between 59 and 62% of the shoppers purchased at least one organic product.
3The probability is 80% that the sample proportion will be between what two symmetrical limits of the population proportion?
4The probability is 99% that the sample proportion will be between what two symmetrical limits of the population proportion? Sixty percent of all grocery shoppers at the local Trader Joes purchase at least one organic product as part of their shopping order. A random sample of 400 shoppers at this Trader Joes is taken.
1What is the sampling distribution of the sample proportion? On what do you base your answer?
2What is the probability that between 59 and 62% of the shoppers purchased at least one organic product.
3The probability is 80% that the sample proportion will be between what two symmetrical limits of the population proportion?
4The probability is 99% that the sample proportion will be between what two symmetrical limits of the population proportion? Sixty percent of all grocery shoppers at the local Trader Joes purchase at least one organic product as part of their shopping order. A random sample of 400 shoppers at this Trader Joes is taken.
1What is the sampling distribution of the sample proportion? On what do you base your answer?
2What is the probability that between 59 and 62% of the shoppers purchased at least one organic product.
3The probability is 80% that the sample proportion will be between what two symmetrical limits of the population proportion?
4The probability is 99% that the sample proportion will be between what two symmetrical limits of the population proportion?

Explanation / Answer

as here n=400 and p=0.6 ; for np =400*0.6 =240>=10 and n(!-p) =160 >=10

therefore it satisfies condition for normal approximation

hence sampling distribution of the sample proportion is approximately normal with

mean proportion =0.60

2)

probability that between 59 and 62% of the shoppers purchased at least one organic product:

3)

here as middle 80% will lie between 10th and 90th percentile;for which z score =1.2816

threfore  symmetrical limits for 80% =mean proportion -/+ z*Std error =0.6 -/+ 1.2816*0.0245 =0.569 ; 0.631

4)

as middle 99% will lie between 0.5th and 99.5th percentile;for which z score =2.5758

threfore  symmetrical limits for 99% =mean proportion -/+ z*Std error =0.6 -/+ 2.5758*0.0245 =0.537 ; 0.663

please revert for any clarificaiton required.

std error of proportion=p=(p*(1-p)/n)= 0.0245

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