SDS 302 – Homework 3 (30 pts) complete solutions correct answers key

SDS 302 – Homework 3 (30 pts) complete solutions correct answers key

Simulation of Sampling Distribution (4 pts)

Go to the simulation application at: http://courses.cns.utexas.edu/ssc302/simulation/sampling.html

Choose a skewed population.

1.      Calculate the mean and standard deviation of the population (top of page). You can use Excel to do this.

2.      According to the Central Limit Theorem, how are the mean and standard deviation of the sampling distribution of the sample means related to the population mean and standard deviation?

3.      Fill out the table below based on the output from the simulator at the bottom of the page. Use a sample size of 10.

Number of samples drawn from the population

Mean of the means

Standard deviation of the means

About 1,000

About 2,000

About 3,000

About 4,000

About 5,000

About 6,000

About 7,000

4.      How do the simulated means and standard errors at the bottom of the screen compare with the population values? Are your values related how you would expect them to be based on Central Limit Theorem? Speculate on why the values you got from the simulation may not be exactly what CLT tells you to expect.

Problems (26 pts)

1.      The net weights (in grams) for a sample of 6 bags of M&Ms were 47.61, 52.06, 48.33, 49.79, 51.68, and 50.43.  Assume that the net weights of bags of M&Ms is normally distributed in the population. (9 pts)

a.       Find the mean and standard deviation for these bags (show your work).

b.      Create a 95% confidence interval for the mean weight of such bags of M&Ms.

c.       Explain in context what your confidence interval tells you about the bag weights of M&Ms.

d.      The manufacturer claims that the net weight of each bag is 47.9 grams, does this seem accurate?

2.      A manufacturer of a new car claims the typical car will average 32 mpg of gasoline. An independent consumer group is skeptical of the claim and thinks the mean gas mileage is significantly different than the 32 claimed. A sample of 23 randomly selected cars produced a mean mpg of 30.15 with a standard deviation of 4.87. Assume that cars’ gas mileage is normally distributed in the population.(9 pts)

a.       The manufactures want to determine if the mean mpg for the cars is significantly different from 32. Write the null and alternative hypotheses for this question.

b.      What are the assumptions of this test? Are the assumptions met?

c.       Perform the appropriate test including the formula that you used, the calculated value of the t-statistic, and the t-critical you compared your t-statistic to using a significance level of .05. Then, state your conclusion to the test.

d.      Now, suppose the researchers had wanted to find evidence specifically that the car obtained less than 32 mpg. Rewrite your null and alternative hypothesis, then repeat the test and state your conclusion.

3.      The mean weight of all 20-year old women is 130 pounds. A random sample of 30 women athletes who are 20 years old showed a sample mean of 126 pounds with a standard deviation of 15 pounds.  Researchers wanted to determine whether the mean weight for 20-year old women athletes is significantly less than 130, using a significance level of 0.05. (8 pts)

a.       Write the null and alternate hypothesis.

b.      Calculate the test statistic and write a conclusion for this question.

c.       Now suppose a sample of 100 women athletes was taken and the same mean (126) and standard deviation (15) was achieved. Repeat the test.

d.      Explain what caused the difference between the outcomes for parts b and c.