# ECO 4000 Statistical Analysis for Economics: Population Variance

### Questions:

-­‐ Read carefully all questions and make sure to answer them rigorously, precisely and concisely.

-­‐ Calculators are allowed as long as the memory is cleared before the test.

### Q1 (25 points)

In the year 2014, a random sample of 96 small companies revealed the following profits losses distribution:

 Profits and Losses (\$1000) Number of companies -25 10 5 12 15 21 30 26 60 19 115 8

1. Calculate the mean and the standard deviation of these company profits and losses. (5points)

In the previous year (2013), a random sample of 150 small companies showed a mean profit of \$28,400 with a standard deviation of \$28,352.

1. Test to see if there has been significant increase in the average profit made by small companies at a 5% significant level. (10points)

Hint: Set the null hypothesis according to the average profit in 2013 is less or equal to the average profit in 2014. (Assume independent samples and same population standard deviations here)

Construct the 90% confidence interval for the mean difference between 2013 and 2014. Is your finding consistent with your answer in (b) (Explain why or why not)? (10points)

### Q 2 (25 points)

With individual lines at its various windows, a post office finds that the standard deviation for normally distributed waiting times for customers on Friday afternoon is 7.2 minutes. To test this claim, the post office experiments with a single, main waiting line and finds that for a random sample of 25 customers, the waiting times for customers have a standard deviation of 3.5 minutes.

With a significance level of 5 percent, conduct an appropriate test on the claim. (State your null hypothesis, test statistic, critical value, decision rule, conclusion) (15 points)

What is the 95% confidence interval for the population variance? (10 points)

### Q3 (25 points)

In conducting her annual review of suppliers, a purchasing agent has collected data on a sample of orders from two of her company’s leading vendors. On average, the 24 shipments from company 1 have arrived 3.4 days after the order was placed, with a standard deviation of 0.4 days. The 30 shipments from company 2 arrived an average of 3.6 days after the order was placed, with a standard deviation of 0.7 days. The average time for shipments to be received is about the same, regardless of supplier, but the purchasing agent is concerned about company 2’s higher variability in shipping time. Using the 0.025 level of significance in a one-tail test, should the purchasing agent conclude that company 2’s higher standard deviation in shipping times is due to something other than chance? (State your null hypothesis, test statistic, critical value, decision rule, conclusion)

### Q4 (25 points)

It has been reported that 8.7% of U.S. households do not own a vehicle, with 33.1% owning 1 vehicle, 38.1% owning 2 vehicles, and 20.1% owning 3 or more vehicles. The data for a random sample of 100 households in a resort community are summarized in the frequency distribution below. At the 0.05 level of significance, can we reject the possibility that the vehicle-ownership distribution in this community differs from that of the nation as a whole? (State your null hypothesis, test statistic, critical value, decision rule, conclusion)

Source: planetforward.org, July 30, 2019

 Number of Vehicles Owned Number of Households 0 20 1 35 2 23 3 or more 22