STA 4210 – Exam 1 – Fall 2012 – PRINT Name _________________________
For all significance tests, use  = 0.05 significance level.
Q.1. A linear regression model is fit by a downtown ice cream vendor, relating Y (units of ice cream sold in a day) to X
(high temperature for the day). She has recorded sales and temperature on a random sample of n = 20 days over her selling
season, and asks you to fit the model. She gives you the following summary statistics, based on the simple linear
regression model, Y = 0 + 1X +  with  ~ N(0,2) and independent:
n
20
X-bar
82
Y-bar
2057
SS_XX
908
SS_YY
617884
SS_XY
22267
p.1.a. Compute the least squares estimates for 1 and 0 and give the fitted equation:
^
b1  _____________ b0  ____________ Y  ___________________________________
p.1.b. The first day had a high temperature of 86 degrees, and sales of 2203 units. Compute the fitted value and residual:
^
^
Y 1  __________________________________ e1  Y1  Y 1  __________________________
p.1.c. Complete the following Analysis of Variance Table:
ANOVA
df
Regression
Residual
Total
SS
546056.5
MS
F*
F(.95)
617884
p.1.d. Based on ANOVA table, what is conclusion regarding H0: 1 = 0 HA: 1 ≠ 0? Reject H0 or Fail to Reject H0
p.1.e. Compute the estimated standard errors of b1 and b0:
s{b1} = ____________________________ s{b0} = _________________________________________________
Q.2. You find a computer output of a regression analysis, relating Y (total cost of a production run of game day t-shirts) to
X (the number of t-shirts produced on the production run). You are given the estimated regression equation, and the
following summary statistics (the production runs ranged from 100 to 400):
n  16
X  250
 X
n
i 1
i
X

2
^
 200, 000 Y  93.50  5.02 X
2
^



 Yi  Y i 
  26615
MSE  i 1 
n2
n
p.2.a. Obtain a 95% Confidence Interval for the mean cost of all production runs of Xh = 200
^
Y h  ___________

^
s Y h  ____________ 95%CI : ___________________________________
p.2.b. Obtain a 95% Prediction Interval for tomorrow’s production run of Xh = 200
^
Y h  ___________ s pred  ____________ 95%PI : ___________________________________
p.2.c. For what size of production run would the widths of the Confidence Intervals and Prediction Intervals be
minimized? Xh = ____________
p.2.d. Obtain a 95% Confidence Interval for the increase in production costs (on average) as the number of t-shirts
produced increases by 1.
Lower Bound = ____________________________________ Upper Bound = _______________________________
Q.3. A simple linear Regression is fit, relating Y to X. You find the following spreadsheet.
X
20
20
30
30
40
40
Y
47
53
60
70
82
78
Y-hat
50
50
65
65
80
80
Y-bar
65
65
65
65
65
65
p.3.a. Compute: Regression Sum of Squares (SSR). Error Sum of Squares (SSE), and Total Sum of Squares (SSTO):
SSR = ______________ dfR = _______ SSE = ___________ dfE = __________ SSTO = __________ dfTO = ____
p.3.b. What proportion of the variation in Y is “explained” by X?
Answer = _____________________________________________
p.3.c. What is the slope of this estimated regression line?
Answer = ______________________________________________
Q.4. True/False (Not related to Q.3):
p.4.a. The width of a prediction interval for an individual case can be narrower than the corresponding confidence interval
for the mean at the same Xh level and same  level, based on the same fitted regression. True or False
p.4.b. The Pearson Correlation Coefficient will always be the same sign as the slope coefficient. True or False
p.4.c. If we obtain a 95% Confidence Interval for 1 to be (-1.2 , 3.6) the P-value for the F-statistic for testing H0: 1 = 0
versus HA: 1 ≠ 0 will be greater than 0.05
True or False
p.4.d. If R2 = 0, all of the observations fall on the fitted equation (with slope not equal to zero) True or False
Q.5. A researcher is interested in the correlation between height (X) and weight (Y) among 12 year old male children. He
selects a random sample of n = 18 male 12-year olds from a school district, and intends on testing H0:  = 0 versus HA: 
≠ 0, where  is the population correlation coefficient.
 X
n
p.5.a. His sample correlation is r 
i 1
 X
n
i 1
i
i

 X Yi  Y
X

  Y  Y 
2 n
 0.60
2
i
i 1
p.5.a.i. Give his test statistic for testing H0:  = 0 versus HA:  ≠ 0:
Test Statistic = __________________________________________________
p.5.a.ii. Reject H0 if the test statistic falls in the range(s) _________________________________________
p.5.b. What is the smallest his sample correlation coefficient can be (in absolute value) for this test to be significant?
Reject H0 if |r| ≥ ________________________________
n
 X
n
Q.6. Show that
i 1
i

 X  0 where X 
X
i 1
n
i