Solutions_Sample Final_STAT 235_1

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Name: ……………………………………….. Stat 235 Elementary Statistics Final: 100 pts Time: 2 hours Instructions: (1) You are allowed to use two handwritten formula/notes sheets (8.5” x 11.5”) and a basic scientific calculator. (2) Unless otherwise indicated, all relevant work must be shown properly in order to receive full credit. Problem # Points Points Earned Part I 5 Part II 1 20 2 16 3 20 4 24 5 15 Total 100

Part I: Mark one of the choices. (5 pts: 1pt each) 1. Population is a. always the same size as the sample. b. the selection of a random sample. c. the collection of all items of interest in a particular study. d. same as a sample. 2. The value which has half of the observations above it and half of the observations below it, is called the a. range. b. median. c. mean. d. mode. 3. The variance of a sample equals 576. The standard deviation of the sample equals a. 13 b. 24 c. 576 d. 28,561 4. The probability of Type I error is a. the confidence level ( 1 − ࠵?) . b. β . c. greater than 1. d. the level of significance ( ࠵?) . 5. The correlation coefficient ( ࠵?) ranges from a. 0 and 1. b. -1 and +1. c. minus infinity and plus infinity. d. 1 and 100.

Problem 1: (20 pts) The proprietor of a clothing store in New York wants to study the average age of his customers. A random sample of 20 customers revealed an average age of 28 years with a standard deviation of 4 years. The study aims to test whether the true average age of customers is significantly greater than 20 years. a) What is the point estimate of the true average age of all customers? (2 pts) b) Construct a 99% confidence interval for µ (4 pts) c) Set up the null and the alternative hypotheses related to the study. (2 pts) d) State the Type I and Type II errors in the context of this study. (3 pts) e) State the appropriate test statistic and its distribution. (3 pts) f) Compute the test statistic. (2 pts)

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g) At α = 0.05 level of significance, test whether the true average age of customers is significantly greater than 20 years. (4 pts) Problem 2: (16 pts) A supermarket chain is redesigning the checkout lanes and is considering two designs. The following summary shows information on customer checkout times at two stores where the two new systems have been installed. System A System B Sample size ࠵? ! =32 ࠵? " =35 Sample mean ࠵? ! ((( = 4.1 minutes ࠵? " ((( = 3.4 minutes Sample standard deviation ࠵? ! = 2.2 minutes ࠵? " = 1.5 minutes Let ࠵? ࠵? be the true population mean checkout time of checkout lanes with System A ࠵? ࠵? be the true population mean checkout time of checkout lanes with System B The management of the supermarket chain is interested in testing whether the mean checkout times of the two systems significantly differ. Perform the hypotheses testing using critical value approach based on a test statistic. a) Formulate the null and alternative hypotheses. (4 pts) b) State the appropriate test statistic and its distribution. (4 pts) c) Compute the test statistic. (4 pts)

d) At α = 0.05 level of significance, state your conclusion in the context of this study. (4 pts) Problem 3: (20 pts) A researcher wants to study the difference between the percentages of users of two toothpastes who will never switch to another toothpaste. In a sample of 500 users of Toothpaste A , 100 said that they will never switch to another toothpaste. In another sample of 400 users of Toothpaste B , 68 said that they will never switch to another toothpaste. Let ࠵? ࠵? be the true population proportion of users of Toothpaste A who will never switch to another toothpaste ࠵? ࠵? be the true population proportion of users of Toothpaste B who will never switch to another toothpaste (a) What is the point estimate for ࠵? ࠵? − ࠵? ࠵? ? (2 pts) (b) Construct a 99% confidence interval for ࠵? ࠵? − ࠵? ࠵? . (4 pts) (c) Interpret the confidence interval computed in (b). (2 pts)

(d) Perform a hypothesis test to test the claim: Proportion of users of Toothpaste A who will never switch to another toothpaste is higher than the proportion of users of Toothpaste B who will never switch to another toothpaste. (i) Formulate the null and alternative hypotheses. (4 pts) Consider the following StatCrunch output related to this study. (i) Identify the P-value from the StatCrunch output assuming the null hypothesis ࠵? 0 is true. (2 pts) (ii) At α=0.01 level of significance, state the general decision rule based on the P-value approach. (2 pts) (iii) At α=0.01 level of significance, state your conclusion in the context of this study. (4 pts)

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Problem 4: (24 pts) (a) Following is a probability distribution of the daily demand of computers at a local store. Demand Probability 0 0.1 1 0.2 2 0.3 3 0.2 4 0.2 (i) What is the probability of selling more than 2 computers per day at this store? (2 pts) (ii) What is the expected daily demand (average daily demand) of this store? (3 pts) (iii) What is the variance and standard deviation of X? (4 pts) (b) Time takes to travel from home to school is normally distributed with ࠵? = 25 minutes and ࠵? = 5 minutes. What is the probability that it will take more than 30 minutes to travel from home to school? (4 pts)

(c) According to a survey, 15% of U.S. adults read e-books. A random sample of 60 U.S. adults is selected. Let ࠵? be the number of adults who read e-books, out of those 60 adults. (i) Is ࠵? a binomial random variable? If yes, identify ࠵? and ࠵? . (3 pts) (ii) Find the probability that number of adults who read e-books is exactly 10 in the sample. (4 pts) (iii) Find the expected value and the standard deviation of X. (4 pts) Problem 5: (15 pts) The following table provides information on prices of digital cameras and number of megapixels of the cameras. Regression analysis is conducted to predict price (Y) of the digital camera based on the megapixels (X) . Initially, the following scatter diagram obtained from StatCrunch. y 180 200 230 120 470 140 180 310 250 110 x 8 10 7 8 15 8 10 12 10 7

What does the scatter diagram indicate about the association between the two variables (positive/negative)? ……………………………………………………………………………………………… (2 pts) To approximate the linear association between Price and Megapixels , the following output was obtained from StatCrunch using simple linear regression analysis.

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(i) Based on the above output, is there a significant linear association between ‘Price’ and ‘Megapixels’? Namely, test ࠵? % : ࠵? ! = 0 vs. ࠵? ! : ࠵? ! ≠ 0 by the P-value approach. a) At α=0.05 level of significance, state the general decision rule based on the P-value approach. (2 pts) b) Identify the P-value from the output and state the conclusion. (2 pts) c) Write the conclusion in plain English in the context of the problem. (2 pts) (ii) (4 pts) Write the estimated regression equation. (iii) (3 pts) Predict the expected price of a digital camera when the number of megapixels is 9.

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