1) ONE SAMPLE TEST OF HYPOTHESIS FOR THE MEAN (σ UNKNOWN)
The owner of a gasoline station wants to study gasoline-purchasing habits of motorists at his station. He selects a random sample of 60 motorists during a certain week, with the following results for the amounts purchased:X-bar = 11.3 gallons, S=3.1 gallons.
1a) if the owner is interested in whether the average gas purchase is different than 10 gallons, what would be the null and alternate hypotheses?
1b) what would be the Type I error?
1c) what would be the Type II error?
1d) for the owner of the gas station, which type of error would be most meaningful?
1e) at the 0.05 level of significance, is there evidence that the population mean purchase was different from 10 gallons?
1f) determine the p-value in e)
2) ONE SAMPLE TEST OF HYPOTHESIS FOR THE PROPORTION
A billing company that collects bills for doctors’ offices in the area is concerned that the percentage of bills being paid by Medicare has risen. Historically, that percentage has been 31%. An examination of 8368 recent bills reveals that 32% of these bills are being paid by Medicare. Using a 0.05 level of confidence, is there evidence of a change in the percent of bills being paid by Medicare?
2a) write the appropriate hypotheses
2b) what would be the Type I error?
2c) what would be the Type II error?
2d) as the billing company, which error would be most important?
2e) what are the assumptions and conditions?
2f) perform the test and find the p-value
2g) state your conclusion
3) TWO-SAMPLE HYPOTHESIS TESTING COMPARING THE MEANS OF TWO INDEPENDENT POPULATIONS
You want to compare the average increase in price of stocks in your portfolio this past year (2014) compared to the year before last (2013). Assume that for this past year (2014) you had a sample of n1 = 8 with the sample mean X-bar1 = $42 and a sample standard deviation of s1 = $4. (You had eight stocks in your portfolio in 2014 with an average price of $42 per share and a sample standard deviation of $4.)
your portfolio contained 15 different stock holdings. This independent sample of n2 = 15 had a sample mean X-bar2 = $34 and a sample standard deviation s2 = $5.
3a) In finding the critical value, how many degrees of freedom are there?
3b) Using the level of significance α = 0.01, what is the critical value for a one-tail test of the hypothesis Ho: Ho: μ1 ≤ μ2 against the alternative H1: μ1 > μ2 ? (You are trying to determine if there is a statistically valid increase in average price of one share of stock in your portfolio from 2013 to 2014.)
3c) In the context of this problem, what would be the Type I error?
3d) In the context of this problem, what would be the Type II error?
3e) As an investor, which error would you be most concerned with?
3f) What is the value of the pooled variance tSTAT test statistic for testing Ho: μ1 ≤ μ2 ?
3g) What is your statistical decision?
4) TWO-SAMPLE HYPOTHESIS TESTING COMPARING PROPORTIONS OF TWO INDEPENDENT POPULATIONS
According to Census estimates, there were about 20 million children between 8 and 12 years old (referred to as tweens) in the United States in 2009. A recent survey was taken of 1,223 of these tweens; 600 boys and 623 girls. Of the 600 boys, 276 reported reading a book for fun, while 324 of the 623 girls stated they had read a book for fun.
4a) What would be appropriate hypotheses (using a two-tail test) to determine if there is a statistical difference between the proportion of boys who read a book for fun and the proportion of girls that read a book for fun?
4b) In the context of this problem, what would be the Type I and Type II errors?
4c) To determine the critical value, how many degrees of freedom are there?
4d) At a 0.05 level of significance, what is the critical value?
4e) What is the value of Z STAT?
4f) What is your answer to the question about whether there is a statistically valid difference between the proportion of boys and girls that read a book for fun?
5) ONE-WAY ANALYSIS OF VARIANCE TEST FOR DIFFERENCES AMONG MORE THAN TWO MEANS
A sporting goods manufacturing company wanted to compare the distance traveled by golf balls produced using four different designs. Ten balls of each design were brought to a local golf course for testing by the club pro. The order that the balls were selected was random and all 40 balls were hit within a short period of time that minimized the effect of the environment. The data collected is summarized in the Excel worksheet that is posted as an attachment.
Question: At the 0.05 level of significance, is there evidence of a difference (using a two-tail test) in the mean distances traveled by the different designs?
In your answer include:
5a) the appropriate hypotheses;
5b) the statistical calculations used for evidence
5c) the p-value (hint: please use either ANOVA in graphing calculator, or stats addin for Windows Excel, or StatsPlus app available (free) online for both Mac); and
5d) your conclusion.