Question 1205138: Based on historical data, your manager believes that 38% of the company's orders come from first-time customers. A random sample of 250 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.35 and 0.45?
Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.
Answer = (Enter your answer as a number accurate to 4 decimal places.)
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
p = 0.38 = population proportion
phat = sample proportion
phat's job is to estimate p
n = 250 = sample size
Compute the z score when phat = 0.35
z = (phat - p)/( sqrt(p*(1-p)/n) )
z = (0.35 - 0.38)/( sqrt(0.38*(1-0.38)/250) )
z = -0.9772
The result is approximate.
Compute the z score when phat = 0.45
z = (phat - p)/( sqrt(p*(1-p)/n) )
z = (0.45 - 0.38)/( sqrt(0.38*(1-0.38)/250) )
z = 2.2802
This value is approximate as well.
The task of finding P(0.35 < phat < 0.45) is roughly the same as P(-0.9772 < z < 2.2802) when p = 0.38 and n = 250.
Use a stats calculator to find that
P(-0.9772 < z < 2.2802) = 0.8245 approximately
I'm not familiar with the wamap stats calculator, so I won't be much help there. Ask your teacher, a classmate, or refer to the wamap help site.
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If you have either a TI83 or TI84 calculator then type in:
normalCDF(-0.9772, 2.2802, 0, 1) and the approximate result is 0.8245 when rounding to four decimal places.
Here are some other alternative calculators- A very user friendly calculator by professor David M Lane. The calculator also displays the shaded diagram which is a nice bonus.
- Use WolframAlpha. Type P(-0.9772 < z < 2.2802) exactly as shown. Make sure the "referring to statistics" option is selected. The WolframAlpha graph is NOT correct so be sure to ignore it (refer to the David M Lane link to see what the graph should look like)
- Use GeoGebra. It can be accessed through either the normal input bar, the CAS mode, or the probability distribution mode. The reference page can be found here
- Use the normDist function on a spreadsheet.
There are many other alternatives that I haven't listed. You can do an internet search to find your favorite one.
Of all of the choices, the 1st option (the David M Lane one) is probably the best for new students.
The only drawback is there doesn't appear to be a way to change the precision.
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Answer: 0.8245 (approximate)
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