Quantitative Methods for MBAs Homework 6 Angelo Mele Solution

Quantitative Methods for MBAs Homework 6 Angelo Mele

This homework tests your understanding of inference.  Each question is worth 20 points.  Please print this paper

and answer the question in the available space.  You can use any software to compute probabilities and/or critical

values of the distributions.  For each problem, explain how you get the solution.

Problem 1

A company has  a plant that produces airbags.  The manager of the company believes that the plant produces at

most a fraction 0.1 of defective airbags.  To check if this is the case, she collects a sample of 500 observations and

calculates a sample proportion of pˆ = 0.12.  The test for the manager’s hypothesis is:

H0: p = 0.1

H1: p > 0.1

and she wants to have a significance level α = 0.05.

Use the following table of probabilities for the Standard Normal Distribution to solve the question.

Standard Normal distribution

P (Z < 1.282) = 0.9

P (Z < 2.576) = 0.995

P (Z < 1.645) = 0.95

P (Z < 1.96) = 0.975

P (Z < 2.326) = 0.99

P (Z < 0.842) = 0.8

1.  (10 points) Compute the bound(s) of the rejection region and the value of the test statistic for this test.

2.  (5 points) Can you reject the null hypothesis?

3.  (5  points) Can you reject  the null  hypothesis  at α =  0.01  significance  level?  Show  your computations  and

explain.

Problem 2

table

x

0

1

2

3

4

5

P (X = x)    0.45    0.3    0.15    0.05    0.03    0.02

1.  (10 points) What is the probability that a car breaks more than twice a year?

2.  (10 points) Suppose you randomly select n  = 20 cars:  what  is the probability that at  least 1  of these  cars

breaks more than twice a year?

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Problem 3

Use the following table to solve the question

Student-t distribution

P (t55> 1.297) = 0.1        P (t55> 1.673) = 0.05

P (t55> 2.004) = 0.025

P (t56> 1.297) = 0.1

P (t57> 1.297) = 0.1

P (t56> 1.673) = 0.05

P (t57> 1.672) = 0.05

P (t56> 2.003) = 0.025

P (t57> 2.002) = 0.025

The height (in centimeters) of high schoolers, indicated by variable X , is normally distributed.  We are interested

in estimating and make inference about the population mean µ.  We collect an i.i.d.  sample of n = 56 observations,

with sample mean x¯ = 166 and sample variance s²= 169.

1.  (10 points) Construct an exact 80% confidence interval for µ

2.  (5 points) Construct an exact 95% confidence interval for µ

3.  (5 points) Construct an exact 90% confidence interval for µ

Problem 4

Use the following table to solve the question

Standard Normal Distribution

P (Z < 1.282) = 0.9           P (Z < 1.645) = 0.95

P (Z < 2.326) = 0.99

P (Z < 2.576) = 0.995

P (Z < 1.96) = 0.975

P (Z < 0.842) = 0.8

In a survey of 128 individuals we calculate a sample proportion of pˆ = 0.53.

1.  Compute a 99% confidence interval for the proportion

2.  (5 points) Compute a 95% confidence interval for the proportion

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3.  (5 points) Compute a 90% confidence interval for the proportion

4.  (5 points) Compute a 80% confidence interval for the proportion

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Problem 5

(Please round all your results to 3 decimal places.) The owner of a bar wants to know the number of beers drank by

the average customer during an evenings.  He collects data from a random sample of n = 33 consumers, obtaining

a sample mean of 6.212 and a variance of 5.047.  He assumes that the sample observations are i.i.d.  draws from a

Normal distribution.

1.  (5 points) Compute an exact 90% confidence interval for the average number of beers consumed in this bar.

2.  (10 points) After observing the sample, the bar owner wants to test if the average number of beers drank in an

evening is 6. Define the rejection region for the test

H0: µ = 6

H1: µ = 6

at a significance level α = 0.05 and compute the value of the test statistic.

3.  (5 points) What is your decision?  Do you reject?

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