Data Analysis Assignment 2
Your submitted document should include the following items. Points will be deducted if the following are not included.
- Type your Name and STAT 250 with your correct section number (e.g. STAT 250-xxx) right justified and thenData Analysis Assignment #2 centered on the top of page 1 below your name the begin your document.
- Number your pages across your entire solutions document.
- Your document should include the ANSWERS ONLY with each answer labeled by its corresponding number and subpart. Keep the answers in order. Do not include the questions in your submitted document.
- Generate all requested graphs and tables using StatCrunch.
- Upload your document onto Blackboard as a Word (docx) file or pdf file using the link provided by your instructor. It is your responsibility for uploading a readable file.
- You may not work with other individuals on this assignment. It is an honor code violation if you do.
Elements of good technical writing:
Use complete and coherent sentences to answer the questions.
Graphs must be appropriately titled and should refer to the context of the question.
Graphical displays must include labels with units if appropriate for each axis.
Units should always be included when referring to numerical values.
When making a comparison you must use comparative language, such as “greater than”, “less than”, or “about the same as.”
Ensure that all graphs and tables appear on one page and are not split across two pages.
Type all mathematical calculations when directed to compute an answer ‘by-hand.’
Pictures of actual handwritten work are not accepted on this assignment.
When writing mathematical expressions into your document you may use either an equation editor or common shortcuts such as: can be written as sqrt(x), can be written as p-hat, can be written as x-bar.
Problem 1: Game Spinner
We will be comparing empirical probabilities (relative frequencies based on an observation of a real-life process) to theoretical probabilities (long-run relative frequency). We will use StatCrunch to simulate this process using a board game spinner. Imagine in a particular game, you picked a card that allowed you to spin the spinnerthree times in a row. We are interested in the total number of spaces moved in thethree consecutive spins. The board game spinner looks like the image below. The spinner is equally likely to land on any given section.
- Build a probability distribution table for the result of a single spin. Present this completed table in your document. You can present the table horizontally or vertically.
- Simulate spinning the spinner 3consecutive times by following the steps below:
Step 1: Open up the data set “Spinner Options”. This contains the value on the 8 spinner locations and shows that each option is equally likely to occur.
Step 2: Click on Applets àSpinner (the second blue box at the top).
Step 3: For the “Labels” option, select Spaces. For the “Weights” option, select Weights.
Step 4: Click Compute! A new window labeled “Spinner experiment” should open. Click the Spin button and StatCrunch will simulate using the spinner once. Click this button 2 more times so that you have a total of 3spins.
Step 5: Click the Analyze button. The data from your 3 spins should now be stored as a column in StatCrunch.
Step 6: Resize the Spinner experiment window so that it contains 1 row of 3 spins. Copy this image of the three spins into your document for your answer to part (b). Below your image provide the number of spaces you obtained for each spin and the total (sum) of the number of spaces you will move with these three consecutive spins.
- In the Spinner experiment window, click the Reset Simulate another 3 spins and store the results in StatCrunch by clicking the Analyze button. Copy an image of the second set of 3 spins into your document. Repeat this process three more times (hitting Reset between each set of three spins) to produce a total of five sets of three spins (including your work from part (b)). Copy the images of each set of three spins each time you run it for your answer to part (c) (i.e. we are expecting four images). Below your images, provide four more sums of the three spins and show your work.
- Now we will simulate 3 consecutive spins 1000 times and find the total number of spaces moved for each. To do this, use the following steps:
Step 1: Under Data à Simulate à select Custom
Step 2: Under “Values in:”, select Spaces. Under “Weights in:”, select Weights.
Step 3: Under “Number of rows and columns:”, enter 3 for Rows and 1000 for Columns.
Step 4: Under Store samples: click the button to the left of Compute for each column (sample) and type sum(Custom) in the box below it.
Step 5: ClickCompute! You should now have 1000recordings of the sum of three spins of the spinner.
Make a properly titled and labeledrelative frequency histogram out of the resulting Sum column. Copy the image of the histogram into your document for your answer to part (d).
- Use your results in part (d) to find the empiricalprobability of moving 4 or less spaces in 3 spins. Show the calculation for this empirical probability and state your probability as a decimal rounded to three decimal places.
- Calculate the theoretical probability of moving 4 or less spaces in 3 spins (i.e. obtaining the sum of the spins to be 4 or less). Use your probability distribution in part (a) and note that spins are independent. (Hint: Recognize that getting a 1 on spin 1, a 2 on spin 2, and a 1 on spin 3 is a different result than getting a 1 on spin 1, a 1 on spin 2, and a 2 on spin 3.) Show how you obtained this probability and provide the answer.
- In a sentence, compare your empirical probability from part (e) to your theoretical probability in part (f).
- How would you expect the empirical probability in part (e) to change if it had been based on a simulation of 10,000 repetitions and why? Answer this question in one to two sentences.
Problem 2: The Weight of a Preschooler
A researcher was interested in analyzing the weight of preschool children. She took a sample of 256 preschool children and measured their weight (in pounds) on their 3rd birthdays. The data was recorded in StatCrunch.
- Use StatCrunch to construct an appropriately titled and labeled relative frequency histogram of the preschooler’s weights. Copy your histogram into your document.
- What is the shape of this distribution? Answer this question in one complete sentence.
- Now overlay your histogram with a normal curve and add a vertical line at the mean.
This can be done by going to Options à Edit in the top left corner of your graph. Inside the histogram graph box, look for Display Options. Next to “Overlay distrib.:” click the arrow next to the word –optional– and select Normal. Then, check the box next to mean under the word “Markers.”Copy and paste this histogram into your document.
- Do you think it is reasonable to use the normal probability model in this case? Answer this question and provide a reason why in one sentence.
- Calculate the sample size, the mean, and the standard deviation of the “Weight” variable using StatCrunch. (Select Stat à Summary Stats à) Copy and paste this table into your document. Round the mean and standard deviation to two decimal places inside this table.
For parts (f) – (j), assume that the distribution of all preschooler weights in the population is normal with the mean and standard deviation found in Part (e) (again use the rounded mean and standard deviation values).Note: you are using the normal distribution for the next three calculations.
- Calculate the probability that a randomly selected preschoolerweighs more than 37 pounds. First, draw a picture with the mean labeled, shade the area representing the desired probability, standardize, and use the Standard Normal Table (Table 2 in your text) to obtain this probability. Please take a picture of your hand drawn sketch and upload it to your Word document (if you do not have this technology, you may use any other method (i.e. Microsoft paint) to sketch the image). You must type the rest of your “by hand” work to earn full credit.
- Verify your answer in part (f) using the StatCrunch Normal calculator (see instructions below) and copy that image into your document. In addition, write one sentence to explain what the probability means in context of the question.
- Use StatCrunch only to calculate the probability that a randomly selected preschooler weighs between 28 and 33 pounds. Copy the normal distribution image from StatCrunch into your document. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.
- A doctor states that preschoolers in the 7th percentile may be deemed underweight. Calculate the maximum weight that places a preschooler in the 7th percentile for weight (thus, calculate the weight where 7% of preschoolers weigh less than this value).Draw a picture (or two), shade area, and use Table 2 to solve this problem. Please take a picture of your hand drawn sketch and upload it to your Word document (if you do not have this technology, you may use any other method (i.e. Microsoft paint) to sketch the image). You must type the rest of your “by hand” work to earn full credit.
- Verify your answer in part (i) using the StatCrunch Normal calculator (see instructions below) and copy that image into your document. In addition, write one sentence to explain what the probability means in context of the question.
Steps to produce StatCrunch normal graphs.
Step 1: Open the calculator by selecting Stat à Calculators àNormal as shown below.
Step 2: Enter the values for the mean and standard deviation found in part 2d into their respective boxes.
Problem 3: Choosing House Color
A real estate broker was interested in the distribution of house color choices in the state of Virginia. It was discovered that 37% of houses are painted using the color brown. A random sample of 19 houses was selected from houses located in Virginia.
- Check if this situation fits the binomial setting. Write four complete sentences addressing each requirement in one sentence each.
- Assuming this situation is a binomial experiment, build the probability distribution in table form in StatCrunch. There are two ways to do this. You may use Data à Compute à Expression and choose the function dbinom. This method relies on you entering the values of the random variable in the first column of your data table. The other way to do this is to use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 19. You may present this table horizontally or vertically and leave the probabilities unrounded.
- Calculate the probability that exactly 7houses in the sample are painted brown using the StatCrunch binomial calculator. Copy this image from StatCrunch into your document. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.
- Calculate the probability that no more than 10 houses are painted brown using the StatCrunch binomial calculator. Again, provide a StatCrunch binomial calculator graph to display your answer. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.
- Calculate the probability that between 6 and 9houses in the sample (inclusive) are painted brown. Show your work using the probability distribution you built in part (b) to answer this question. Then, verify it with a StatCrunch binomial calculator graph and include this image in your document as well. Finally, once you obtain your answer, write one sentence to explain what the probability means in context of the question.
- Calculate the mean and standard deviation of this probability distribution. Show your work using the binomial mean and standard deviation formulas and provide your answers in your document. (No need to use StatCrunch for this part).
Problem 4: Building a Sampling Distribution
We will use the Sampling Distribution applet in StatCrunch to investigate properties of the sampling distribution of the proportion of credit card chips that are read correctly the first time. A few years ago, credit and debit card chip readers became more widespread. The chip in the card encrypts information to increase data security when making transactions. At a particular store, customers complained that the newly installed readers do not read their chips on the first try. The store owner collected data and determined that the probability that the chip reader worked the first time was 0.89.
Under Applets, open the Sampling distribution applet (box shown below). First, select Binary for the population. Next, to the right of “p:”, enter 0.89, which is the proportion of credit card chips that are read correctly the first time they are entered. Then click on Compute! See image below.
- Once the applet box is opened, enter 10 in the box to the right of the words “sample size” in the right middle of the applet box window (see image below). Then, at the top of the applet, click “1 time.” Watch the resulting animation. When the sample is completed, copy and paste the entire applet box (using options à copy) into your document.
- Click Reset at the top of the applet. Then, click the “1000 times” to take 1000 samples of size 10. Copy and paste the applet image into your document.
- Describe the shape of the Sample Proportions graph at the bottom of your image from part (b) in one sentence.
- Why do you think that this graph does not have an approximately normal shape? Use the Central Limit Theorem large sample size condition to answer this question in one sentence. Explicitly show these calculations.
- Click Reset at the top of the applet. Type 100 in the sample size box. Then, click the “1000 times” to take 1000 samples of size 100. Copy and paste the applet image into your document.
- Describe the shape of the Sample Proportions graph at the bottom of your image from part (e) in one sentence.
- Why do you think that this graph from part (f) has the shape you described? Use the Central Limit Theorem large sample size condition to answer this question in one sentence. Explicitly show these calculations.
- Using the image in part (e), write the values you obtained for the mean (in green) and the standard deviation (in blue). These values are found in the bottom right box labeled “Sample Prop. of 1s.”
- Compare the mean value (in green, found in part (h)) to the known population proportion in one sentence.
- Now calculate the standard error of the sample proportion using p = 0.89 and n = 100 by hand. Type your “by-hand” and round your answer to four decimal places.
- Compare the value in part (j) to the standard deviation (in blue) you obtained in part (h) in one sentence.
- Use the sampling distribution defined by the Central Limit Theorem to calculate the probability that from a sample of 100 students at least 86% of credit card chips are read the first time they are used (use p = 0.89 and the standard error found in part (j)). First, draw a picture with the mean labeled, shade the area representing the desired probability, standardize, and use the Standard Normal Table (Table 2 in your text) to obtain this probability. Please take a picture of your hand drawn sketch and upload it to your Word document (if you do not have this technology, you may use any other method (i.e. Microsoft paint) to sketch the image). You must type the rest of your “by hand” work (e.g. the formula to calculate the z-value) to earn full credit.
- Provide a StatCrunch Normal graph to verify the work in part (l) and interpret the resulting probability in context.
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