Lab 3 – Binomial Distribution



Math 10 MPS: Lab 4 – Continuous Random Variables & Central Limit Theorem

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Math 10 MPS Lab04. Make sure you copy all members in the group.

For this lab, open Minitab file Lab4Data.MPJ

1. Simulate a Binomial/Sample Proportion Random Variable

The Binomial random variable is described by two parameters, the number of independent trials (n) and the probability of success on a single trial (p). In this simulation, we will model the following binomial random Variable:

In a poll conducted January 2015, 72% of American adults put protecting freedom of speech ahead of not offending others. Assume this is the true proportion. You sample 64 American adults. Let X be the number in the sample who put protecting freedom of speech ahead of not offending others.

a. Use the column heading Binomial Sim to save data and simulate 1000 trials (use the menu item CALC>RANDOM DATA, choose Binomial and set the correct parameters.)

b. Using the command STAT>BASIC STATISTICS>GRAPHICAL SUMMARY to calculate the sample mean, sample median and sample standard deviation of the simulated data as well as a box plot and histogram and paste the output here.

c. Describe the shape of the histogram.

d. Calculate n(p) and n(1-p). Is the shape of the histogram what you would expect based on these results? Explain.

e. Calculate [pic]and σ =[pic]. Compare the sample mean and sample standard deviation to the population values.

2. Simulate a Normal Random Variable

The Normal random variable is described by two parameters, the expected value µ the population standard deviation σ. The curve is bell-shaped and frequently occurs in nature. In this simulation, we will model the cooking time for popcorn which follows a Normal random variable with µ =4.2 minutes and σ = 0.6 minutes.

a. Use the column heading Normal Sim to save data and simulate 1000 trials (use the menu item CALC>RANDOM DATA and choose Normal.)

b. Using the command STAT>BASIC STATISTICS>GRAPHICAL SUMMARY to calculate the sample mean, sample median and sample standard deviation of the simulated data as well as a box plot and histogram and paste the output here.

c. Describe the shape of the histogram. Does it appear to match the bell-shape of the population probability graph shown above?

d. Identify the minimum and maximum values. Determine the Z-score of each. Do these values seem to be extreme outliers?

e. Compare the sample mean, median and standard deviation to the population values.

Central Limit Theorem for the Sample Mean (Questions 3, 4, 5)

The lifetime of optical scanning drives follows a skewed distribution with µ =100 weeks and σ =100. The five columns labeled “CLT for mean n= ##” represent 1000 simulated random samples of 1, 5, 10, 30, and 100 from this population.

3. Make dot plots of all 5 sample sizes using the Multiple Y's Simple option and paste the result here.

a. As the sample size changes, describe the change in center.

b. As the sample size changes, describe the change in spread.

c. As the sample size changes, describe the change in shape.

4. Using the command STAT>DISPLAY DESCRIPTIVE STATISTICS, determine the mean and standard deviation for each of the five groups. Paste the results here.

a. As the sample size increases, describe the change in mean.

b. As the sample size increases, describe the change in standard deviation.

5. What you have observed are the three important parts of the Central Limit Theorem for the distribution of the sample mean [pic] In your own words, describe these three important parts.

Central Limit Theorem for the Sample Proportion (Questions 6, 7, 8, 9)

6. First, let's identify the population proportion. If you roll a six-sided die, what proportion would you expect to come up 6? Write this as a fraction and a decimal.

The columns labeled “CLT for p n=##” represent 1000 trials of rolling a six-sided die 5, 10, 50, 100 or 500 times, and determining the sample proportion of times a 6 was rolled. We know that the sample proportion is a random variable that we want to investigate. We are going to use Minitab to make some dot plots for this data to determine the center, spread and shape of the distribution of the sample proportion.

7. Make a Simple dotplot of the sample proportion for sample size n=5. Paste the graph here.

a. What is the center of the data?

b. What is the spread of the data (use range)?

c. Describe the shape of the data.

8. Make a Simple dotplot of the sample proportion for sample size n=50. Paste the graph here.

a. What is the center of the data?

b. What is the spread of the data (use range)?

c. Describe the shape of the data.

9. Now we are going to make a Multiple dotplot for all 5 sample sizes using the Multiple Y's Simple option. Paste the graph here and answer the 3 questions about the distribution of the sample proportion.

a. How has the center changed as the sample size increased?

b. How has the spread (range) changed as the sample size increased?

c. How has the shape changed as the sample size increased?

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