Module 6 Homework: Sampling Distribution and Confidence Intervals

Homework Exercises:

Part I – Sampling Distribution (10 pts)

For this Part I exercise, enter your answers in your Word (or PDF) document, and show your work in the Excel document provided in the dropbox ( hw-module6-simple random samples.xls ).

The purpose of this exercise is to illustrate the concept of a sampling distribution, using a very small sample from a very small population. The population is 10 scholarship players currently on a women’s basketball team (imagine DePaul’s women’s basketball team). The 10 players have been labeled with the integers 0 to 9. For each player, the total amount of time spent (in minutes) on Facebook during the last week is recorded in the table below and provided in the Excel document.

The population parameter of interest is the average amount of time on Facebook.

1. Compute the mean ‘ time ’ for the 10 players in the population. This is the population mean µ.

1. Show your work.
2. Enter population mean µ here: _____________

1. Use Excel ’s RAND () function to draw a simple random sample (SRS) of size n=3 drawn from this population of players.
   1. Use the Excel document provided for this assignment. Draw the first SRS in the first worksheet tab labeled "Sample 1".
   2. See the video provided on the Content page in D2L for Module 6 on how to use the RAND() function in Excel. An example is also provided in your textbook on pages 95-96.
2. Compute the sample mean from the three players’ times that were in your SRS. This sample mean is the statistic that is an estimate of µ.

• For your first SRS, s how your work on the first worksheet tab, labeled "Sample 1" in the Excel document provided for this assignment. In other words, show which players were included in the sample, and the calculation of the sample mean, based on these players’ times.

1. Repeat Steps 2 and 3 above 9 more times – each time, drawing an SRS, recording the 3 players’ times, and computing the sample mean.
   1. For each SRS, use a different worksheet tab in the Excel document provided for this assignment. Note : there should be a total of 10 SRSs, each on a separate worksheet tab, labeled samples 1 through 10.

• ( Note : You may include the same player’s time in more than one sample, as long as the player is randomly selected for the sample.)

1. For each SPS drawn (i.e., generated; created), compute the sample mean \[\bar{x}\] for the three players’ times in the sample. Include the sample mean calculation in the corresponding sample worksheet in the Excel document.

1. Make a histogram of the 10 values of \[\bar{x}\] . In doing so, you are approximating the sampling distribution of \[\bar{x}\] . \[\]
   1. Use can use either SPSS or Excel to create the histogram. Tip: SPSS is quicker, but you may find it interesting to see how to do more in Excel, such as create a histogram. If so, here is a 4-min video on creating a histogram in Excel: https://www.youtube.com/watch?v=RyxPp22x9PU . It’s simple to do. The key difference between creating the histogram in Excel versus SPSS is you will need to define the ranges of the data into classes of equal width (as you did for Module 1 HW); Excel refers to these classes as ‘bins’. If interested in creating your histogram in Excel, watch the movie above; otherwise, use SPSS.
   2. Briefly describe the shape of your histogram (e.g., approximately Normal or not; skewed or not, and if so, in what direction; unimodal, bi-modal, or no distinct peaks)
   3. [copy and paste your graph here]
2. Recall the population mean µ that you computed in step 1 above. Is the center of your histogram in from Step 5 close to µ? ______________(yes/no)

Briefly explain your answer (e.g., in 1, no more than 2 sentences).

Part II – Confidence Intervals

Problem 1. (3 pts)

You want to rent an unfurnished two-bedroom apartment in Chicago next year. Imagine that the mean monthly rent for a random sample of 10 apartments advertised in a local online news site is $1600. Assume that the monthly rents in Chicago follow a Normal distribution with a standard deviation of $475.

• Find a 95% confidence interval for the mean monthly rent for unfurnished 2-bedroom apartments available in Chicago. Show your work (i.e., clearly show how your answers were derived).

Problem 2 . (7 pts)

Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset. The mpg values for a random sample of 20 of these records are provided in the data file mpg.xls (provided in the D2L dropbox for this HW assignment).

Suppose that the standard deviation is known to be \[\sigma \] = 3.5mpg.

1. What is the \({{\sigma }_{{\bar{X}}}}\) , the standard deviation of \[\bar{x}\] ? _________________ .

• Show your work.

1. Examine the data for skewness and other signs of non-Normality using any two methods of your choice. (For example, see the methods you used in the Module 5 homework.)

• As applicable to your chosen methods, copy and paste your graphs here, and/or show your numerical calculations.

1. Do you think it is reasonable to construct a confidence interval based on the Normal distribution? _________(yes/no).

• Briefly explain your answer in reference to your results shown for step 2 above.

1. Give a 95% confidence interval for µ, the mean miles per gallon for this vehicle._______________

• Show your work.

1. Interpret the confidence interval in your results for step 4 above.___________________________

______________________

Price: $21.81

Solution: The downloadable solution consists of 9 pages, 1281 words and 2 charts.
Deliverable: Word Document