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Statistics 251 –Examination Three – Summer 2007
NAME : __________________________ STUDENT ID : _________________
I pledge on my honor that I have not given or received any unauthorized assistance on this examination. ____________________________ ______________________________ Signature Date
Setting One (Questions 1 through 4: 20 Points): The Princeton Review regularly rates
American colleges and universities on a variety of elements including how many hours students study each week and perceived quality of the institution. This information is widely cited and often provides criteria by which students, their parents, or guardians choose an institution to attend. Suppose that a sample of 500 public colleges and universities was selected and each was categorized according to the two traits with the results recorded in the table below. Conduct the appropriate test of hypothesis at the α = .05 level of significance. STUDY HOURS
PERCEIVED QUALITY OF INSTITUTION
Low to Medium High Prestigious Totals
Low 100 48 2 150 Medium 200 42 8 250 High 50 30 20 100
Totals 350 120 30
1. Provide H 0 and HA in sentences specific to the problem (4 points) H 0 : **HA:
- Provide the complete decision rule (4 points)** DR:
3. What is calculated value of the test statistic? χ
2 calc = __________ (6 points)
4. Provide a reasonable and context specific interpretation of results (6 points) Statistics 251-Summer 2007 - Examination Three
SETTING TWO: (Questions 5 through 9: 20 points) A recent survey gathered responses
from a random sample of 500 married couples. Among the goals of the work were to identify issues regarded as most important in a marriage. Family counseling specialists suspect that the proportion of couples citing “shared views on child raising” will be regarded as less important now than 20 years ago with many of these specialists citing smaller family sizes and an increase in the proportion of couples choosing to remain childless. Twenty years ago 42% of married couples cited “shared views on child raising” as the single most important issue in their marriage. Test the appropriate hypothesis at the α = .10 level of significance and construct the appropriate 90% confidence interval. To do this suppose that 150 of the 500 couples sampled cited “shared views on child raising” as the single most important issue in their marriage.
5. Provide H 0 and HA in sentence form with words specific to the issue (4 points). H 0 : **HA:
- Provide the complete decision rule (4 points).** **DR:
- What is the calculated value of the test statistic? Calculated Value = _________ (4 pts)
- Construct the appropriate 90% confidence interval.** ________ < < ________ **(4 pts)
- Provide a reasonable and context specific interpretation (4 points)** Statistics 251-Summer 2007 - Examination Three
Setting Four: (Questions 15 through 22: 40 points): Refer back the beginning of “setting
three” for a description of I
3
. Suppose that I
3 has a process for embossing diplomas on parchment paper. The quality of the embossing is related to a number of factors including the quality of the parchment, printing speed, width of the paper, and the pressure at which the ink is applied to the
paper. I^3 believes that, in particular, Y = quality of the embossing (rated from “1” = low to “10” =
exceptional) is related to X = print speed (which is rated in units per second and ranges from 1 to
5). I^3 is able to control or set print speed, but embossing quality is believed to result from print
speed and other factors. Data follows for a sample of 20 diplomas: Y = Quality 2 2 3 4 3 4 5 5 5 6 6 7 6 8 8 9 8 9 10 10 ∑Y = 120 ∑Y^2 = 844 X = Speed 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 ∑X = 60 ∑X^2 = 220 ∑XY = 426 A plot of the data appears below: 1 2 3 4 5 2 3 4 5 6 7 8 9 10 Speed Quality
15. Provide the means, variances, standard deviations, and covariance where X = Print Speed and Y = Quality. (10 points total – 2 for each of the variances and covariance). Mean of X= __________ Mean of Y = __________ Variance of X = __________ Variance of Y = __________ Standard Deviation of X = __________ Standard Deviation of Y = __________ Covariance of X and Y = __________ 16. What are the estimated SLOPE (β 1 ) and INTERCEPT (β 0 ) values? (4 pts – 2 each) Estimated Slope, β 1 = _______ Estimated Intercept, β 0 = _______ Statistics 251-Summer 2007 - Examination Three
17. Use the regression equation to determine the estimated QUALITY (that is, Y-hat) when PRINT SPEED is set at 4. (3 points) Y-hat = __________ 18. On the plot of the data provided on the previous page, ACCURATELY plot the regression line. In the space below, provide the coordinates of the two points that you are using to plot the line? (4 points – 2pts for the plot and 1 each for the two points used) Point 1 coordinates = ____________ Point 2 coordinates = ____________ 19. What are the values of the CORRELATION COEFFIENT (r ) and COEFFICIENT OF DETERMINATION (r^2 ). Also, provide a context specific interpretation of r^2. Correlation Coefficient = __________ (2 points) Coefficient of Determination = __________ (1 point) Interpretation (2 points): 20. Complete the analysis of variance (ANOVA) table below. Use α = .05 to obtain a critical value of F. (4 points) ANALYSIS OF VARIANCE TABLE Source of Variation Degrees of Freedom Sum of Squares Mean Squares Fcalc Fcrit Regression (e.g. Print Speed) Residual Total 21. Construct and interpret the 95% confidence interval for the slope. (5 points … 3 for the interval and 2 for the interpretation) ______________ < β 1 < _______________ Interpretation: 22. Construct and interpret the 95% confidence interval for the ESTIMATED QUALITY when PRINT SPEED is 4. (5 points … 3 for the interval and 2 for the interpretation) ______________ < < _______________ Interpretation: Statistics 251-Summer 2007 - Examination Three
