Statistics Lab 3.1 PDF

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LAB 3.1 Statistics 200: Lab Activity for Section 3.1 Sampling Distributions – Learning Objectives:      

Distinguish between a population parameter and a sample statistic, recognizing that a parameter is fixed while a statistic varies from sample to sample Compute an estimate for a parameter using an appropriate statistic from a sample Recognize that a sampling distribution shows how sample statistics tend to vary Recognize that statistics from random samples tend to be centered at the population parameter Estimate the standard error of a statistic from its sampling distribution Explain how sample size affects a sampling distribution

Activity 0: This lab uses StatKey. Open the StatKey guide available on Canvas. Activity 1: Creating a sampling distribution. To complete this activity, you will use a virtual box of small colored beads. As a class, we are going to build a sampling distribution for the proportion of blue beads in the box. 1. Open the virtual box (open the link in the description above). What is the population? Blue and red beads 2. By just looking at the picture of the virtual box, what do you think the proportion of blue beads is? .50 3. Now take a random sample of n=30 beads from the box. Each student must take her or his own sample, even if you’re working in a group! To take a sample just click the draw a sample button. You may want to save a screen shot of your sample for future reference. Using this sample, what is the sample proportion? Your answer should be correctly rounded to two decimal places, i.e. 0.xx. .67 4. Now what do you think the proportion of blue beads in the box is? Has your answer changed from question 2? .67, yes 5. Enter your sample proportion using the Canvas link in the submodule for this section. It is very important that you do this so that you and your classmates can create a sampling distribution to discuss in the last activity.

6. Discuss with the other people in your breakout room about how different your sample proportions were.

9/18/20

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LAB 3.1

Activity 2: Which parameter, which statistic? 1. A random sample of n=461 smartphone users in the US in January 2015 found that 355 of them have downloaded at least one app1. a. Give notation for the parameter of interest, and define the parameter in this context. p, parameter is the proportion of smartphone users that have downloaded at least one app b. Give notation for the quantity that gives the best estimate for this parameter, and give its value. P (hat) = 0.77 2. Of the n=355 smartphone users in question 1 who had downloaded an app, the average number of apps downloaded was 19.7. a. Give notation for the parameter of interest, and define the parameter in this context. b. Give notation for the quantity that gives the best estimate for this parameter, and give its value. 3. For questions 1 and 2 above, what would we have to do to calculate the parameters exactly?

4. Florida has over 7,700 lakes2. We wish to estimate the correlation between the pH levels of all Florida lakes and the mercury levels of fish in the lakes. The correlation between these two variables for a sample of n=53 lakes is -0.575. a. Give notation for the parameter of interest, and define the parameter in this context. Rho, the correlation between the pH level and mercury lakes b. Give notation for the quantity that gives the best estimate for this parameter, and give its value. R = -0.575 5. A study wants to investigate whether or not regular exercise leads to a lower resting heart rate. A random sample of Stat 200 students were classified as either “no” or “yes” with regard to regular exercise. For the “no” group (group 1) the average resting heart rate was 72.0 beats/minute and for the “yes” group (group 2) the average resting heart rate was 65.2 beats/minute. a. Give notation for the parameter of interest, and define the parameter in this context. Mu 1 – mu 2 b. Give notation for the quantity that gives the best estimate for this parameter, and give its value. X bar 1 – x bar 2, 6.8 Activity 3: Beads revisited: our sample results and implications of sample size 1 Olmstead, K., and Atkinson, M. “Apps Permissions in the Google Play Store,” pewresearch.org, Novermber 10, 2015. 2 www.stateofflorida.com/florquicfac.html

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© - Pennsylvania State University

LAB 3.1 Observe the dataset of sample proportions generated by you and your classmates. You and can find the dataset following the link on Canvas in the submodule for this section. Copy and paste the column of sample proportions into Minitab and create a dotplot of the sample proportions. 1. Observe this dotplot of sample proportions: a. What does each dot represent? A sample proportion b. What is the shape of this sampling distribution? Where is the center? Symmetric/bell, .5 c. Recall that the standard error for a sample statistic is the standard deviation of the sampling distribution. Use Minitab to find the standard error of your estimate from Activity 1. (Hint: find the standard deviation of the class dataset of sample proportions ). Sd= .0889 2. What is your best guess for the population proportion of blue beads in that bead box? .55 3. Assuming that your answer for Question 2 above is in fact the correct population proportion, let’s investigate how the sampling distribution would change if we had taken larger or smaller samples. Instead of taking more samples from the bead box, we will use StatKey to simulate this process. The program will randomly generate many samples (and sample proportions) from a pre-determined population. a. Generate a single sample of size n=100 from a population with the proportion you specified in question 2. What was the sample proportion? .50 Now generate at least 2000 sample proportions and observe the sampling distribution dotplot. b. What does each dot represent? Sample propotion c. Where is the center of the sampling distribution? .55 d. What is the standard error of the sample proportion with size n=100? 0.051 4. Finally, create a sampling distribution from the same population (same proportion) but with samples of size n=200, with at least 2000 samples. a. What is the standard error? 0.034 b. Where is the center of the distribution? 0.05 5. Use your answers from the questions above to summarize how increasing the sample size affects the standard error. Decreases standard error How does it affect the center of the sampling distribution? No change

9/18/20

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