# Random variables, Expected Values, Chi-Square and Levels of Significance

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Question 8: In business, time is money. A local entrepreneur wants to start a regional 24/7 package delivery service for businesses along the front range. A key planning assumption is that, with Colorado Springs as the collection and distribution hub, package delivery times throughout this region will follow a Normal Probability Distribution. Using US Postal Service data and other information, an analyst constructs the following projected package delivery times (in hours) for the front range region:

Hours Frequency

1 up to 3* 4

3 up to 5 12

5 up to 7 21

7 up to 9 30

9 up to 11 25

11 up to 13 8

* Recall that "up to 3" doesn't include 3.

Given the hypotheses: H0: The package delivery time is approximately Normal**

H1: The package delivery time is not approximately Normal

** "Normal" means that the Normal Probability Distribution is followed.

a. What is the random variable for this problem.

b. Using Expected Value, find the mean and standard deviation for the information given.

c. Using the 0.01 level of significance and a Chi Square Goodness of Fit test, state your conclusion as to whether the package delivery times, in hours, are Normally distributed (follow the Normal pattern).

keywords: chi-squared, squared

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##### Solution Summary

Random variables, Expected Values, Chi-Square and Levels of Significance are investigated.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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