Sarah is the marketing manager of a retail store in the space. In order to desig

Question

Sarah is the marketing manager of a retail store in the space. In order to design a royalty plan, she needs to know what is the probability, denoted by p, that a randomly selected customer spends more than $100 at the store. According to a sales record, 25 out of 80 randomly selected customers spent more than $100. Answer the following questions and help Sarah construct 99% confidence interval for p.
Must USE MiniTab where needed

1) What are the conditions that Sarah has to check to conclude that a normal distribution can provide a good approximate for the distribution of p?

2) If the conditions specified in Part 1 hold, clearly specify the normal distribution that approximates the distribution of p.

Use the normal distribution that you obtained in Part 2, and answer the following.

3) Calculate the critical values.

4) Calculate the confidence interval limits.

5) Schematically draw the distribution of p and show the following items on the plot: a) a point estimate of p b) critical values c) area for the surface under the curve to the right of the upper limit and to the left of the lower limit.

Explanation / Answer

1 & 2.
Points to pass for normal approximation:
a. experiment consistes of a sequence of n identical trials
b. only 2 outcomes are possible on each trail, success or failure
c. trials are independent & below conditions should satisfy
n*p>5, 80*0.3125> 5 => 25>5
n*(1-p)>5, 80*0.6875> 5 => 55>5
can use normal approximation

3.
Za/2 = Z-table value
level of significance, = 0.01
from standard normal table, two tailed z /2 =2.576

4.
TRADITIONAL METHOD
given that,
possibile chances (x)=25
sample size(n)=80
success rate ( p )= x/n = 0.3125
I.
sample proportion = 0.3125
standard error = Sqrt ( (0.3125*0.6875) /80) )
= 0.0518
II.
margin of error = Z a/2 * (stanadard error)
where,

margin of error = 2.576 * 0.0518
= 0.1335
III.
CI = [ p ± margin of error ]
confidence interval = [0.3125 ± 0.1335]
= [ 0.179 , 0.446]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=25
sample size(n)=80
success rate ( p )= x/n = 0.3125
CI = confidence interval
confidence interval = [ 0.3125 ± 2.576 * Sqrt ( (0.3125*0.6875) /80) ) ]
= [0.3125 - 2.576 * Sqrt ( (0.3125*0.6875) /80) , 0.3125 + 2.576 * Sqrt ( (0.3125*0.6875) /80) ]
= [0.179 , 0.446]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 99% sure that the interval [ 0.179 , 0.446] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 99% of these intervals will contains the true population proportion

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