1. Discuss why it is important to collect a random sample from the population to provide a good estimate for the population statistics.

Them more representative and informative a sample is the better the estimate for the population statistic will be. The estimate for the population statistic is only as good as the information available in the sample. If the sample is a poor reflection of the population then the estimate will not likely be accurate.

  1. What type of sampling method would you choose to use to estimate the unemployment rate for the entire United States? Explain what advantages this method would have given the U.S. population.

Given the geographic and socio-demographic variation in the U.S. population a stratified random sample of the labor force would provide the best estimate of the unemployment rate.

  1. The population mean income for recent graduates with business degrees is $54,000 with a population standard deviation of $14,000. Assume a random sample of 125 recent graduates is collected. Use this information to answer the following questions.
  1. What is the expected value of the sample mean? Interpret this value.

The expected value of the sample mean is 54000. If we were able to collect many samples and calcualte the average income for recent graduates for each sample. The expected value of those averages would be equal to the population mean 54000.

  1. What is the standard deviation for the sample mean? Interpret this value.

The standard deviation for the smple mean is 1252.198. The standard deviation of the sample mean represents the variation in the estimate for the mean.

  1. What is the probability that a sample mean will be less than $52,000?

The probability that the sample mean will be less than $52,000 is 0.055

  1. What is the probability that a sample mean will be greater than $55,000?

The probability that the sample will be greater than $55,000 is 0.212

  1. What upper and lower bounds would put 5% in the outside of the distribution. Let \(\alpha\) = 0.05.

56454.263

51545.737

  1. A marketing firm collected data on the number of people who made an online purchase after a promotional email was sent out to their customer list. From the population, 24% of the people on the list made a purchase after the promotional email. Assume a sample of 500 out of a total 1000 people on the list are surveyed. Use this information to answer the following questions.
  1. What is the expected value of the sample proportion? Interpret this value.

The expected value of the sample proportion is 24%.

  1. Is it necessary to use the finite correction factor for this example? Explain.

Yes. The sample constitutes 50% of the overall population. It is necessary to use the finite correction factor in this case.

  1. Calculate the standard deviation for the sample proportion. Use the finite correction factor if necessary. Interpret this value.

The standard deviation for the sample proportion is 0.014.

  1. In New York City it is generally accepted that 1% of workers commute to work by bike. A sample of 300 workers in New York are surveyed on their transportation method to work. Can the sampling distribution for the proportion of people who commute to work by bike be assumed to be normal? Explain your answer using the properties of the sampling proportion.

With a sample of 300 workers you would only have around 3 observations of people biking to work. This is not enough instances to meet the normality assumption.