3/22/99 252y9921 ECO252 QBA2 Name

SECOND HOUR EXAM Hour of Class Registered (Circle)

March 23, 1999 MWF TR 10 12 12:30 2:00 night

I. (14 points) Do all the following. Use diagrams! Show your work!

(Why do too many of you still believe that a probability can be negative?)

1.

2.

3.

4.

5. (The Cumulative probability)

6. A symmetrical interval about the mean with 68% probability.

We want two points , so that. From the

diagram, if we replace x by z, . The closest we can come is

. So (1.00 is acceptable and something in

between even better), and , or –1.91 to 15.91.

check:

7.

We want a point , so that. From the diagram, if we replace

x by z, . The closest we can come is

(though 1.69 or something between the two is acceptable). So , and

, or 22.3.

check:

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II. (6 points-2 point penalty for not trying part a.) Show your work!

A new software package has been designed to help system analysts design information systems. We wish to see if there is a significant difference between the time required to develop a system using the new technology and using the current technology. Two independent samples are taken. The results are shown below. Assume that the data represent independent samples taken from populations with the normal distribution.

Current Technology New Software

300276

280222

344310

385338

372200

360302

288

I will be happy to nominate everyone who had trouble dealing with for the Bourbon prize for inflexibility. Samples only need to be the same size when data is paired and there is no reason to believe that this data is paired!

  1. Compute , the standard deviation for time required to design a system using the new software. (3) Note that .
  2. Do a two-sided confidence interval for the difference between the means for the two technologies

. (2) Indicate the following: What assumptions did you make about the variances of the

populations from which the samples were taken? Is there a significant difference between the means

(You must tell why)? (1)

Solution:


b. Solution: Assume equal variances.From Table 3 of the Syllabus Supplement:

Interval for / Confidence Interval / Hypotheses
/ Test Ratio
/ Critical Value
Difference
between Two
Means (
unknown,
variances
assumed equal) /

/ /
/

.

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or –23.3 to 139.3. This interval includes 0. Since the interval includes zero, we can say that there is no significant difference between the means. Or if the null hypothesis is or or , we cannot reject it.

b. Alternate Solution: Assume unequal variances. From Table 3 of the Syllabus Supplement:

Interval for / Confidence Interval / Hypotheses / Test Ratio / Critical Value
Difference
between Two
Means(
unknown,
variances
assumed
unequal) /
/
Same as
/ /

, so use 9 degrees of freedom.

, so or –28.4 to 144.4. Conclusion is the same as for a.

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III. Do at least 3 of the following 5 Problems (at least 10 each) (or do sections adding to at least 30 points - Anything extra you do helps, and grades wrap around) . Show your work! State and where applicable.

  1. A firm is currently contracting with firm 1 for delivery of raw materials. It will stay with firm 1 unless it can demonstrate that delivery times for firm 2 are faster. The statistics from the two samples are below. Assume that the parent populations are Normal.

Firm 1 Firm 2

  1. Test to see if the mean for population 1 is larger than the mean for population 2.

(It’s incredible that 2/3 of you couldn’t write down “” here!)

(i)State your hypotheses in terms of and . (1)

(ii)Do the test assuming that . (3)

(iii)Do the test assuming that . (4)

(iv)On the basis of these tests would you change supplier? (why?) (1)

  1. Test the variances for equality. (3)

Solution: See previous pages for formulas

  1. (i) or or .

(ii)

(You still can’t do a 2-sided test for a 1-sided Hypothesis!)

Test Ratio: This is above 2.37.

or Critical Value:.

1.5 lies above this value.

or Confidence Interval:

This interval does not include 0. In all cases reject .

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(iii)

, so use 77 degrees of freedom.

, so or –28.4 to 144.4.

Test Ratio: This is above 2.37.

or Critical Value:.

1.5 lies above this value.

or Confidence Interval:

This interval does not include 0. In all cases reject .

(iv) As usual, rejection of a hypothesis calls for action. Since , we

have shown that firm 2 is faster.

b. . Since the second ratio is below 1, the only one we need check is . Since is larger than our ratio, do not reject .

Solution continues in252z9921

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