Posted: January 12th, 2023

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◼ ABC Enterprises is a growing company in which employees must frequently travel for business presentations in the metro area of the city where the company is located. Currently, each employee is allotted 35 dollars each day (total) to cover meals when they are giving these presentations. Employees have complained that this is not enough, especially when they give more than one presentation in a day. A survey was taken to get recommendations for a new allotment. Although not all responded, 58% of the employees did respond.

This week, we are going to use the Central Limit Theorem. We are going to pretend that our data set represents the entire population of ABC Company. It doesn’t, but we are going to pretend that it does.

1. What is the mean dollar amount that is recommended by the population of employees?

2. Now, we are going take 3 samples of 10 dollar amounts from the spreadsheet and average (find the mean of) them. Open the spreadsheet: see attached december 2022 A.xlsx

Go to the random number generator at https://www.google.com/search?q=random+number+generator&oq=random+number&aqs=chrome.0.0j69i57j0l6.3428j0j7&sourceid=chrome&ie=UTF-8

(or any other random number generator)

Put in the minimum number of 5 (this aligns with the cell that begins your list of numbers).

Put in the maximum number of 67 (this aligns with the cell that ends your list of numbers).

Hit “generate.”

The number given tells you what cell to go to. For example, if I got a 48 on the random number generator, I would put my cursor in cell 48 and record the number that is there. For example, if there were a “75” in that cell on the Excel spreadsheet, I would write down 75. Continue to hit the random number generator 9 more times. Record the other 9 values in the cells. Do not worry about repeats. Provide your list of the 10 values from the cells.

Find the mean of those ten values in Excel. Write down your sample mean.

Repeat this exercise two more times. You will have a total of three sample means.

3. Comment on your three means of the samples. Were they the same? How did they compare to the mean of the population? Were the samples representative of the population? Why or why not?

4. Now, take all 30 numbers that you got from your three samples, and find the mean of all 30. How close is it to your population mean?

5. If you spent all day taking samples of 10 from the population and finding the mean of each sample and then made a histogram of these samples means, it would form a bell shaped curve. How does this relate to the Central Limit Theorem?

6. Make sure to upload your spreadsheet.

◼ABC Enterprises is a growing company in which employees must frequently travel for business presentations in the metro area of the city where the company is located. Currently, each employee is allotted 35 dollars each day (total) to cover meals when they are giving these presentations. Employees have complained that this is not enough, especially when they give more than one presentation in a day. A survey was taken to get recommendations for a new allotment. Although not all responded, 58% of the employees did respond.

Open the Excel Spreadsheet: see attached

Weekly Assignment Spreadsheet December 2022 fulljk.xlsx

You have a sample of employees at ABC Enterprises on the spreadsheet, and this is a sample of the population. We want to use this sample to learn about the entire population of workers at the company. We are going to treat the population as if it is not a finite population.

Annie is a company employee who grew up with 9 older brothers. She noticed that they tended to eat more than she did, and she assumed that this characteristic applied to the general population. In other words, she believes that males eat more than females. She wondered if the proportion of male employees in the company was impacting the survey results.

Answer these questions:

What proportion of the sample are male?

Why can’t we say that the proportion of male employees in the entire company is the same as the proportion of male employees in the sample?

Section 9.4 discussed the Confidence Intervals for a population proportion. We are going to determine a 95% confidence interval for the proportion of employees in the entire company that are male. What is the z sub alpha over two value for a 95% confidence interval? What is the value of n?

Use the formula to create the confidence interval for the proportion of male employees in the entire company. Remember order of operations (you should be starting by working what is under the square root sign). What is your confidence interval?

Why is there a range for the answer instead of one single value like there was in your first response in this post?

Calculate the 99% confidence interval for the proportion of the population who are male.

Compare the range of the 95% confidence interval to the range of the 99% confidence interval. What happened, and why?

How would you respond to Annie’s concern about the proportion of males in the company affecting the dollar amount?

◼ 1.What is the area under a standard normal curve? Why is it that number?

2. A statistics student was about to hand in some homework when his friend pointed at his answers and said, “You may want to recheck those.” The friend saw this:

The probability of the event is

a. -1/3

b. 1.5

c. 0.27

d. 0

e. 1

f. Which answer(s) was the friend concerned about, and why?

3.Examine the bell-shaped curves below and answer the questions.

The picture above is the Standard Normal Distribution. Where is the mean on that curve?

The picture above is the Standard Normal Distribution. Where is the median on that curve?

The picture above is the Standard Normal Distribution. Where is the mode on that curve?

Look at the picture of the different curves in the picture above. What happens to the height of the curve as the standard deviation increases?

4.Explain how to compute the z value. Start with the formula. Tell us what each variable means. Then, explain what the z value tells us about the value of the random variable. Use the numbers in 5a (below) to aid in your explanation.

5.Let x be a normally distributed random variable with µ=32 and σ=2.17. Find the z value for each of the following observed values of x: Round to TWO decimal places if needed. Show your work.

a.X = 35

b.X = 30

c.X = 34.17

d. X = 28.2

e. X = 32.15

6. In #5,

a. which response (a, b, c, d, e) is the most uncommon due to how far it is from the mean?

b. which response(s) are within one standard deviation of the mean? How do you know?

7.If the random variable z has a standard normal distribution, sketch (insert your photos of your sketches) and find each of the following probabilities using a z table. You may need to reference an online z table.

a.P( -1.66 < z <0.15)

b.P( z – 2.04)

Begin your sketch with a drawing of the standard normal distribution pictured below:

Then draw a vertical line(s) for your z value(s). Last, shade in the areas according to the > or < signs. Your drawings might look similar to those below, except they would have the z scores from -3 to +3 across the x axis.

*** don’t forget to use the z table to then find the probability!

8.Weekly demand at a grocery store for a brand of breakfast cereal is normally distributed with a mean of 300 boxes and a standard deviation of 12 boxes. What is the probability that the weekly demand is:

a.280 boxes or less?

b.More than 305 boxes?

c.Between 270 and 301 boxes?

** Show your problem set up and work for #8

◼A company surveyed its employees to determine what the new daily meal allotment dollar amount should be. Of the 100 employees, 74 responded. Pat’s supervisor asked what the recommended dollar amount should be. Pat could take the mean of the 74 responses or calculate a confidence interval about the mean using those 74 responses. Which action of the two (don’t go outside these choices) would give the supervisor a better idea of what the true mean is, and why?

2.You are catching fish in a pond and measuring the length of fish. You want to know the mean length of the population of fish, but you also know you can never be sure you have caught and measured all the fish, so you decide to use confidence intervals. You catch fish 10 at a time, measure their lengths, and create 90% confidence intervals each time. Assume you have data from 100 samples of fish taken from the pond and have calculated the 90% confidence interval for each of these samples.

a. Why will you have different confidence intervals for each of your trials?

b. How many of those 100 confidence intervals will contain the true population mean? Why?

3.You take one of your samples of fish and calculate the 90% confidence interval for the mean length of the fish, and you compare it to the 99% confidence interval for the same sample. Did the range of the interval increase or decrease, and why?

4. Your samples usually contained 10 fish. You have been calculating the 90% confidence interval for the mean length of the fish for each sample. Let’s say that your next sample contained 50 fish. If you calculated the 90% confidence interval for the sample of 50 fish, would the range of its 90% confidence interval from this sample of 50 be larger or smaller than the range of the 90% confidence interval when the sample size was 10 fish? Why?

5.Suppose that for a sample size of n=222, we find that the sample mean is 15 inches. Assuming that the standard deviation = 0.8 inches, calculate the confidence intervals for the population mean µ with the following confidence levels. Show your problem set up (work) and calculations.

a.90%

b.92%

c.95%

d.97%

e.99%

6.Compare the shape of the t curve to the shape of the z curve by finding pictures of them online. What are their similarities and differences?

7. Locate a flowchart online which helps you determine when to use t and when to use z in your calculations. The flow chart should include knowledge of population standard deviation and sample size. Embed it in this assignment. Cite your source. Explain how to use the flowchart given this scenario:

You have a sample size of 197 and you know the population standard deviation. Will you be using z or t?

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