Statistics / Mr. Hansen
4/20/2004


Check one:

                 Name: ____________KEY____________
c Regular time (36 minutes)
c Extended time (54 minutes)

Test on Chapters 13 and 14, Version 1

Instructions and Scoring.

  • Show all work and answers on separate sheets.
  • Show as much work as time provides. Justify all steps except for algebraic simplification or area calculations. For full credit, you should show the details of s.e. and m.o.e. calculations to the extent possible.
  • You will not earn full credit for a correct answer without adequate justification, or a correct answer without correct notation.

 

 

 

 

1.

The owner of a bakery claims that her loaves have weight that is normally distributed with mean 16.2 oz. and s.d. 0.8 oz.

 

 

(a)

Assuming that this is true, fill in the following chart for an SRS of 250 loaves. Give all answers to 3 decimal places, and verify that the percentages add to 100% except for possible rounding errors.

 

 

 

Weight

% expected (probability)

Number of loaves expected

 

less than 15 oz.
15.0 to 15.5 oz.
15.5 to 16 oz.
16 to 16.5 oz.
16.5 to 17 oz.
more than 17 oz.

6.681 %
12.398 %
21.051 %
24.488 %
19.518 %
15.866 %

16.702
30.995
52.627
61.219
48.794
39.664

 

 

(b)

What do the expected number of loaves add up to? 250

 

 

(c)

Here are some observations for the SRS of 250 loaves.

 

 

 

Weight

Observed frequency

 

 

less than 15 oz.
15.0 to 15.5 oz.
15.5 to 16 oz.
16 to 16.5 oz.
16.5 to 17 oz.
more than 17 oz.

8%
12%
20%
26%
20%
14%

 

 

 

 

Conduct a goodness-of-fit test, showing all steps. Is there evidence to refute the baker’s initial claim?

 

 

 

Let p1 thru p6 = expected frequencies for 6 bins as shown based on assumption of N(16.2, .8).
H0: Bread weights follow N(16.2, .8) [or you could say p1 thru p6 are as shown above]
Ha: Bread weights do not follow N(16.2, .8) [or you could say p1 thru p6 are not all as shown above]

Assumptions:
 ü SRS
 ü All expected counts ³ 1 [all ³ 16, in fact]
 ü No more than 20% of counts < 5 [none, in fact]

Test statistic:
 c2 = S (obs. – exp.)2/exp. = (.08 · 250 – 16.702)2/16.702 + (.12 · 250 – 30.995)2/30.995 + . . . = 1.626

p = .898

Conclusion: There is no evidence (c2 = 1.626, df = 5, p = .898) that the bakery owner’s claim is false. [The largest contribution to c2 is in the first cell, meaning that the relative deviation from the expected count was greatest for the loaves under 15 oz. There are many more of these than predicted by the N(16.2, .8) distribution, but overall, still not enough to refute the bakery owner’s claim. The observed deviations could plausibly be explained by chance.]

 

2.

Researchers are interested in the relationship between GPA and time spent playing video games for students at Bali High School. Specifically, they wish to know whether video gaming time can be used as a predictor of GPA. The following data came from an SRS of students:

 

 

 

 

GPA

Video Gaming Time (Hrs./Wk.)

 

 

 

2.4
2.6
2.6
2.85
3.0
3.3
3.45
3.7
3.9

6
5
7.5
9
9.5
8
10.2
10.4
11

 

 

 

 

(a)

Describe the relationship between the variables, using words that indicate the context of the problem. Provide at least one diagram and at least one piece of quantitative evidence to support your claim.

(b)

Using video game time as a predictor of GPA, state the line of best fit as a mathematical model in which variables are defined.

(c)

Interpret the slope in part (b) in the context of the problem, using words such as “video gaming time” and “GPA.” Your answer should make sense to someone who has studied little or no statistics.

(d)

Compute a 95% confidence interval for the slope of the linear regression model.

(e)

Perform a significance test for the proposition that the true slope is positive. Show all steps.

 

 

(a)

There is a strong positive linear relationship (r = 0.855) between hours/wk. spent w/ video games and GPA. A residual plot [should be shown here] reveals no obvious pattern to doubt the validity of the linear fit.

 

 

(b)

yhat = 1.219885 + .2195957x, where x = hrs./wk. of video gaming and yhat = predicted GPA

 

 

(c)

slope = b1 = .2195957 Þ For each additional hr./wk. of video gaming, the linear model predicts an increase of .2195957 GPA units, more than a fifth of a letter grade. [For full credit, you must use the word “model” or “predicts.”]

 

 

(d)

sb1 = b1 / t = .2195957/4.370378 = .050246
C.I. = est. ± m.o.e. = b1 ± (t*)(s.e.) = .2195957 ± 2.365(.050246) = (.1008, .3384)

We are 95% confident that the true slope (b) of the regression model is between .1008 and .3384.

 

 

(e)

Let b = true LSRL slope

H0: b = 0
Ha: b > 0

Assumptions:
 ü random sample of data points
 ü true assoc. appears to be linear since resid. plot shows no pattern [must show resid. plot]
 ü resids. do not vary w/ change in x [must show resid. plot]
 X resids. do not appear to be normally distributed [must show stemplot, histogram, or NQP];
 however, for a sample this small, we cannot say for sure

Proceed w/ caution since normality assumption appears to be violated.

Test statistic: t = 4.370378 by calc.

p = .0016 by calc. (1-sided)

Conclusion: Except for the problem w/ non-normality of resids., there is strong evidence (t = 4.37, df = 7, p < .002) that the true value of the slope is positive.

 

 

(e)

ALTERNATE METHOD

 

 

 

[Proceed as before with assumptions. Then, instead of stating t and p, simply refer to the C.I. previously calculated in (d).]

Conclusion: Except for the problem w/ non-normality of resids., we can conclude w/ more than 95% confidence that the true LSRL slope is positive. Reason: 95% C.I. of (.1008, .3384) does not include 0; this C.I. has positive vals. only.