Samples of Standard Normal Random Variables

Samples of Standard Normal Random Variables 

1. Use a computer to generate a sample of 50 standard normal random variables. Generate a histogram for your sample. Set the random number seed to the sum of your group members birth dates, display the seed, and a list of the numbers. For example if your three group members were born on the 5th, 12th, and 21st, then use the seed 5 + 12 + 21 = 38. 

2. Calculate the sample mean. 

3. Calculate the sample standard deviation. 

4. We know that population μ = 0 and σ = 1. Use this information to construct a 95% confidence interval for sample mean. 

5. Does this confidence interval contain μ. 

Sample Means of Real Data - NHL Goals 

Download the NHL dataset (attach file)

1. Calculate the population mean of goals. 

2. Calculate the population standard deviation of goals. 

3. Select a random sample of 75 players from the population. One way to do this in excel is to add a column of randomly generated uniform(0, 1) variables, sort the sheet by this column, and use the first 75 observations for goals. Create a histogram of goals for your sample. 

4. Calculate the sample mean. 

5. Calculate the sample standard deviation. 

6. Calculate a 95% confidence interval for the sample mean. Does this interval contain μ? 

7. Construct the smallest possible confidence interval for your sample mean that contains μ. (Hint, do this by using the formula for a confidence interval and setting μ = to either the upper or lower limit, depending on whether μ is above or below x ̄, and solve for the critical value. 

Sample Mean Hypothesis Testing 

Imagine that you have a portfolio of stocks. You want to compare the performance of your stocks to the market as a whole. You learn that the population mean,μ, of stock returns is 7%, and that the population standard deviation, σ, is 2%. 

1. Imagine that you have a portfolio of 10 stocks, that have a sample mean of 14%. Construct a 90% confidence interval. 

2. What is the probability that the sample mean is equal to the population mean? 

3. Imagine that you have a portfolio of 10 stocks, that have a sample mean of 2% returns. Construct a 95% confidence interval. 

4. What is the probability that the sample mean is equal to the population mean? 

5. Imagine that you have a portfolio of 10 stocks, that have a sample mean of 8%, what is the probability that the sample mean is equal to the population mean? 

Sample Proportion Hypothesis Testing 

Individuals are often classified as being morbidly obese if they are 100 pounds overweight. Imagine that the population proportion of morbidly obese individuals in the country as a whole is 2%. 

1. How large does n need to be to use the standard normal distribution?
2. You obtain information on 500 individuals from a town and calculate the sample proportion to be 4%. 3. Calculate a 90% confidence interval for sample proportion?
4. Does the confidence interval include the population proportion?
5. What is the probability that the sample proportion is equal to the population proportion?