SOLUTION: Based on historical data, your manager believes that 28% of the company's orders come from first-time customers. A random sample of 151 orders will be used to estimate the proporti

Algebra ->  Probability-and-statistics -> SOLUTION: Based on historical data, your manager believes that 28% of the company's orders come from first-time customers. A random sample of 151 orders will be used to estimate the proporti      Log On


   

Question 1201801: Based on historical data, your manager believes that 28% of the company's orders come from first-time customers. A random sample of 151 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.22 and 0.43?
Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 0.9497 (approximate)

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Explanation:

I'm not familiar with wamap, so I won't be much help there. But I can help with the rest.

Given info:
"Based on historical data, your manager believes that 28% of the company's orders come from first-time customers"
This leads to p = 0.28 as the population proportion. We convert from percent form to decimal form. Move the decimal 2 spots to the left.
The value of p is always between 0 and 1 inclusive, ie 0+%3C=+p+%3C=+1


More given info:
"A random sample of 151 orders will be used to estimate the proportion of first-time-customers."
meaning
n = 151 = sample size


Summary so far
p = 0.28 = population proportion
n = 151 = sample size

Now we look at the phat distribution.
Sometimes it is spelled out as "p-hat"
It's named as such because it is the letter p with a small hat on top.
phat =

That font is probably a bit distorted, so here's a link that might offer a better image
https://tex.stackexchange.com/questions/250466/vector-with-boldface-and-a-hat-on-top

The phat is the sample proportion.
phat's job is to estimate p.

The phat distribution has these 2 properties
  • mean = p
  • standard deviation = standard error = sqrt(p*(1-p)/n)
I'll use "SE" as the abbreviation of "standard error".

In this case, we have
  • mean = 0.28
  • SE = sqrt(0.28*(1-0.28)/151) = 0.036539 approximately
We can use a normal distribution with these two parameters.

The goal is to compute P(0.22 < phat < 0.43)
It gives the probability the sample proportion phat is between 0.22 and 0.43

Let's compute the z score when phat = 0.22
z = (phat - p)/SE
z = (0.22 - 0.28)/0.036539
z = -1.64208106406853
z = -1.6421

Repeat for phat = 0.43
z = (phat - p)/SE
z = (0.43 - 0.28)/0.036539
z = 4.10520266017132
z = 4.1052

The task of computing
P(0.22 < phat < 0.43)
is roughly equivalent to
P(-1.6421 < z < 4.1052)

Now use a calculator such as this one
https://davidmlane.com/normal.html
or something like a TI83 or TI84

Or you could use a table like this
https://www.ztable.net/
The drawback with the table is you must round each z score to 2 decimal places. Thereby losing accuracy.

I'll go with the calculator approach. I'll assume the calculator you're using is similar to the one in the link.

You should find that
P(-1.6421 < z < 4.1052) = 0.9497
which leads back to
P(0.22 < phat < 0.43) = 0.9497
This value is approximate.

If we randomly picked phat values from the phat distribution, then there's roughly a 94.97% chance of getting a phat in the interval 0.22 < phat < 0.43
phat = sample proportion