Directions: a. b. c. For the following problems, include the following information in your answer: Determine if the problem is either a test of hypothesis, a confidence interval or something else and specify the 'key words' found in the problem that demonstrate your choice. Determine the procedure name and parameters involved for each problem (use the Stat 200 Formulas and Techniques Summary document, which is on the next page after problem #6.) Specify the 'key words' found in the problem that lead you to this choice. If the problem is a hypothesis test, indicated if it has a lower-tail, upper-tail, or two-tail alternative hypothesis as well as the test statistic formula for the test. If the problem is a confidence interval then indicate the formula used for the margin of error. You DO NOT need to do the ACTUAL test of hypothesis, confidence interval, etc., just answer parts a, b, and c for the problem. 1. The following data were obtained from a sample of construction companies who are all building 10-story structures. The survey asked how many full-time workers are employed on the building project. 152 163 142 110 198 125 173 212 200 165 315 Historically, 10-story structures are built on a budget which has a forecasted average employment of 175 full-time workers. At the 10% significance level, is there evidence that the mean number of full-time employees has increased? 2. A new method of storing snap peas is believed to retain more ascorbic acid than an old method. In an experiment, snap peas were harvested under uniform conditions and frozen in 62 equal-size packages. Thirty five of these packages were randomly selected and stored according to the new method and the other thirty two packages were stored by the old method. Subsequently, ascorbic acid determinations were made, and the following summary statistics were calculated. New method Old method Mean 449 451 St. dev. 19 45 Sample size 35 32 Do these data substantiate the claim that more ascorbic acid is retained under the old method of storing at the 5 % level of significance? 3. A new method of storing snap peas is believed to retain more ascorbic acid than an old method. In an experiment, snap peas were harvested under uniform conditions and frozen in 62 equal-size packages. Thirty five of these packages were randomly selected and stored according to the new method and the other thirty two packages were stored by the old method. Subsequently, ascorbic acid determinations were made, and the following summary statistics were calculated. New method Old method Mean 449 440 St. dev. 19 45 Sample size 35 32 4. Estimate the difference in ascorbic acid concentrations between the two groups with 95% confidence. A study is to be made of the relative effectiveness of two kinds of cough medicines in increasing sleep. Six people with colds are given medicine A the first night and medicine B the second night. Their hours of sleep each night are recorded. Medicine A Medicine B 1 4.8 3.9 Subject 2 4.1 4.2 3 5.8 5.0 4 4.9 4.2 5 5.3 5.4 6 7.4 7.0 Is there a significant difference in the amount of sleep when switching from medicine A to medicine B at the 5% significance level? 5. Government officials near a nuclear power plant have been working on an emergency preparedness program. In 2004, a random sample of 405 community residents was obtained and 275 said they were confident they would be notified quickly of a radioactive incident. In 2005, after the program started, a random sample of 378 community residents was obtained, and 210 felt they were confident they would be notified quickly of a radioactive incident. At the 5% significance level, is there evidence that the true proportion of residents who were confident they would be notified quickly of a radioactive incident in 2005 declined from that of 2004? 6. The following table shows the amount spent on business lunches for twelve senior executives at a large corporation. Is there evidence at a .05 significance level that there is a difference in amount spent determined by corporation title? Chief Financial Officer $49 $58 $44 $42 Chief Executive Officer $50 $55 $46 $61 Chief Administrative Office $38 $45 $32 $43 Summary Table for Statistical Techniques Inference Parameter Statistic Type of Data 1 Estimating a mean One population mean sample mean x numerical 2 Test about a mean One population mean sample mean x numerical 3 Estimating a proportion 4 Test about a proportion One population proportion p One population proportion p sample proportion p sample proportion p Analysis Minitab Command What is the average weight of adults? What is the average cholesterol level of adult females? 1-sample t-interval s x t * n Stat >Basic statistics >1-sample t Is the average GPA of juniors at Penn State higher than 3.0? Is the average Winter temperature in State College less than 42 F? What is the proportion of males in the world? What is the proportion of students that smoke? Ho : = o Ha : o or Ha : > o or Ha : < o Is the proportion of females different from 0.5? Is the proportion of students who fail Stat200 less than 0.1? H o : p = po Ha : p po or Ha : p > po or Ha : p < po Examples categorical (binary) categorical (binary) The one sample t test: x 0 t s n 1-proportion Z-interval p z * p (1 p) n The one proportion Z-test: Stat >Basic statistics >1-sample t Stat >Basic statistics >1-sample proportion Stat >Basic statistics >1-sample proportion Conditions data approximately normal or have a large sample size (n 30) data approximately normal or have a large sample size (n 30) n p 10 and n (1- p ) 10 n po 10 and n (1-po ) 10 z Inference Parameter Statistic Type of Data Examples 5 Estimating the difference of two means difference in two population means 1-2 difference in two sample means numerical x1 x 2 6 7 8 Test to compare two means difference in two population means 1-2 difference in two sample means numerical x1 x 2 Estimating a mean with paired data mean of paired difference D sample mean of difference Test about a mean with paired data mean of paired difference sample mean of difference numerical d numerical p p0 p0 (1 p0 ) n Analysis Minitab Command How different are the mean GPAs of two-sample t-interval Stat males and females? >Basic * How many fewer ( x1 x 2 ) t s.e.( x1 x 2statistics ) colds do vitamin C >2-sample t takers get, on See text, page 445, for average, than non the s.e. of the difference vitamin C takers? Do the mean pulse rates of exercisers and non-exercisers differ? Is the mean EDS score for dropouts greater than the mean EDS score for graduates? What is the difference in pulse rates, on the average, before and after exercise? H o : 1 = 2 Ha : 1 2 or Ha : 1 > 2 or Ha : 1 < 2 Is the difference in IQ of pairs of twins zero? H o : D = 0 Ha : D 0 or Ha : D > 0 or Ha : D < 0 The two sample t test: ( x x2 ) 0 t 1 s.e.( x1 x2 ) Conditions Stat >Basic statistics >2-sample t independent samples from the two populations data in each sample are about normal or large samples (ni 30) independent samples from the two populations data in each sample are about normal or large samples (ni 30) See text, page 445, for the s.e. of the difference paired t-interval d t * sd n Stat >Basic statistics >Paired t Stat >Basic statistics differences approximately normal or have a large number of pairs (n 30) differences approximately normal or D Inference Parameter d Statistic Type of Data Examples 9 Estimating the difference of two proportions difference in two population proportions p1-p2 difference in two sample proportions p1 p 2 categorical (binary) 10 Test to compare two proportions 11 Relationship in a 2-way table difference in two population proportions p1-p2 relationship between two categorical variables or difference in two or more difference in two sample proportions p1 p 2 Are the pulse rates of people higher after exercise? categorical (binary) the observed categorical counts in a two-way table How different are the percentages of male and female smokers? How different are the percentages of upper- and lowerclass binge drinkers? Is the percentage of males with lung cancer higher than the percentage of females with lung cancer? Are the percentages of upper- and lowerclass binge drinkers different? Is there a relationship between smoking and lung cancer? Do the proportions of students in each class who smoke t d0 sd n Analysis >Paired t Minitab Command have a large number of pairs (n 30) Conditions two-proportions Z-interval ( p1 p 2 ) z * s.e.( p1 p 2 ) Stat >Basic statistics >2 proportions Stat >Basic statistics >2 proportions independent samples from the two populations n p 10 and (1-n) p 10 for each sample See notes for s.e. formula H o : p 1 = p2 Ha : p1 p2 or Ha : p1 > p2 or Ha : p1 < p2 The two proportion z test: z ( p1 p 2 ) 0 s.e.( p1 p 2 ) independent samples from the two populations n p 10 and (1-n) p 10 for each sample See notes for s.e. formula Ho : The two variables are not related Ha : The two variables are related The chi-square statistic: Stat >Tables >CrossTabu lation >ChiSquare analysis for For summarized Data: Stat> all expected counts should be greater than 1 at least 80% of the cells should have an expected count greater than 5 2 population proportions differ? all cells Inference 12 Test about a slope Parameter slope of the population regression line 1 Statistic sample estimate of the slope b1 Type of Data (Observed Expected) 2 Expected Examples Analysis H o : 1 = 0 Ha : 1 0 or Ha : 1 > 0 or Ha : 1 < 0 numerical Is there a linear relationship between The t test with n-2 degrees of height and weight of freedom: a person? b 0 t Tables> ChiSquare. Minitab Command Stat >Regression >Regression 1 13 Population Sample means means of the of the k k populations numerical populations 1,2,....., ,k x1,x2,....., ,xk Is there a difference Ha : not all the means are equal between the mean The F test for one-way GPA of Freshman, ANOVA: Sophomore, Junior and Senior classes? MSTR F MSE s.e.(b1 ) Ho : 1=2= =k Test to compare several means Conditions Stat >ANOVA >Oneway the form of the equation that links the two variables must be correct the error terms are normally distributed the errors terms have equal variances the error terms are independent of each other each population is normally distributed independent samples from the k populations equal population standard deviations