Econ 2810 Intro to Statistics and Computing In Economics Spring 2016 Final Exam Review Final Exam Time and Date: 10:20-12:20pm, May 10, 2016 True or False (15 points) 1. The standard error is equal to s . n 2. As the number of degrees of freedom increase, the t distribution gets closer and closer to the normal distribution. 3. The 95% confidence interval for the mean of a normal population with known standard deviation is ( ) X z0.05 . n1 4. For a two sided test, if | t |> t0.05 , then we can reject the null hypothesis at the 10% level of significance. 5. To use the t test statistic for statistical inference, the population variance ( 2 ) is assumed to be known. Multiple Choice Questions (15 points) 1. The one-sample t statistic from a sample of n = 21 observations for the two-sided test of H0 : = 60 H1 : = 60 has the value t = 1.98. Based on this information: a. we would reject the null hypothesis at = 0.10. b. 0.025 < P value < 0.05 c. we would reject the null hypothesis at = 0.05 d. none of the above 2. The costs of four new cellular phones are $160, $215, $99 and $195. The standard error of the mean of these four cellular phones is: a. 50.86 b. 167.25 c. 12.71 d. 25.43 [Next Page] 1 2 3. We test the null hypothesis H0 : = 10 and the alternative H1 : = 10 for a normal population with = 4. A random sample of 16 observations is drawn from the population and we find the sample mean of these observations is X = 12. The P value is approximately a. 0.0228 b. 0.0456 c. 0.4886 d. 0.9772 4. If the degrees of freedom = 10, then t0.001 is a. 3.169 b. 4.297 c. 2.764 d. 4.144 5. The pooled two sample t statistic can be used if a. The two populations have unequal variances and equal sample sizes b. The two populations have unequal variances and unequal sample sizes c. The two populations have equal variances and equal sample sizes d. None of the above are correct Short Answer (40 points) 1. A researcher is studying the failure rate of restaurants. She selects a random sample of 200 restaurants in large cities that opened within the last year. Following up on these restaurants, she finds that 80 had failed within five years. Calculate a 99% confidence interval for the proportion of restaurants that fail within five years. 2. Based on a sample of the salaries of professors at a major university, you have performed a multiple regression relating salary to years of service and gender. The estimated multiple linear regression model is: Salary = $45,000 + $3000(Years) + $4000(Gender) + $1000[(Years)(Gender)] where Gender = 1 if the professor is male and Gender = 0 if the professor is female. a. Using the above multiple linear regression equation, calculate the average salary of male professors with three years of experience. [Next Page] 3 3. Consider the following two way ANOVA table. Find the values for a, b, c, d, e, f. Source Brand df 3 SS 3410 MS 1136.8 F d P value 0.000 Club 1 a 46443.9 e 0.000 Brand*Club 3 105.2 b f 0.260 Error 24 590.4 24.6 Total 31 c 4. Independent random samples are selected from two normal populations. Information for the 2 samples are in the table below: n X S Population 1 14 7.1 2.3 Population 2 11 8.4 2.9 If you wanted to test the means are, in other words you are testing the following hypotheses: H0 : 21 22 = 0 H1 : 21 22 = 0 a. Setting = 0.05, do you have enough statistical evidence to reject the null hypothesis? Why or why not? b. Now you want to test whether the population variances are equal. Setting = 0.05, do you have enough statistical evidence to reject that the population variances are equal? Long Answer (30 points) 1. In a completely randomized design experiment, 10 individuals were randomly chosen for each of three treatment groups and a quantity was measured for each unit within each group. In the first steps of testing whether the means of the three groups are the same, the sum of squares for treatments was calculated to be 3110 and the sum of squares for error was calculated to be 27000 a. Complete the following ANOVA table Source Treatment df SS MS Error Total b. Suppose you want to test whether the means are identical. Write down the null and alternative hypotheses c. Setting = 0.05, can you reject the null hypothesis you constructed in part b)? Why or why not? [Next Page] 4 2. A random sample of 79 companies from the Forbes 500 list (which actually consists of nearly 800 companies) was selected, and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear regression model was used Profitsi = 0 + 1 (Sales)i + i where the deviations i were assumed to be independent and normally distributed. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: R2 = 0.662 s = 466.2 Variable Constant Sales Parameter Est. 176.644 0.092498 Std. Err. of Parameter Est. 61.16 0.0075 a. What is the value of the intercept? b. Construct a 90% confidence interval for the slope 1 c. Suppose the researchers test the hypotheses: H 0 : 1 = 0 H 1 : 1 > 0 Can you reject the null hypothesis if = 0.01? d. Suppose we wish to predict the mean profits (in hundreds of thousands of dollars) for all companies that had sales (in hundreds of thousands of dollars) of 500. Given that sales is 500, what is the predicted value for mean profits