Mgmt 671 Review of Statisticalconcepts

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A NOTE ON HYPOTHESIS TESTING

|Significance Level |One-Sided Test |Two-Sided Test | |0.10 |1.285 |1.645 | |0.05 |1.645 |1.960 | |0.01 |2.33 |2.575 |

Part A. Single-Sample Inference

1. Test Statistic for the Population Mean ((): Large Sample Test

H0:(=(0
H1:(((0

Test statistic:

[pic], where (0=hypothesized value of (

Note that:
Sample mean=[pic]
Mean of the sample mean: E([pic])=(
Standard error of[pic]=[pic]

Example 1

As part of a survey to determine the extent of required in-cabin storage capacity, a researcher needs to test the null hypothesis that the average weight of carry-on baggage per person is μ 0 = 12 pounds, versus the alternative hypothesis that the average weight is not 12 pounds. The analyst wants to test the null hypothesis at α = 0.05. The data collected for this study are:

n=144;[pic]= 14.6;s = 7.8;For α = 0.05, critical values of z are ±1.96

H0: μ = 12[two-tailed test]
H1: μ ≠ 12

[pic]

Since the test statistic falls in the upper rejection region, H0 is rejected, and we may conclude that the average amount of carry-on baggage is more than 12 pounds.

Example 2

The EPA sets limits on the concentrations of pollutants emitted by various industries. Suppose that the upper allowable limit on the emission of vinyl chloride is set at an average of 55 ppm within a range of two miles around the plant emitting this chemical. To check compliance with this rule, the EPA collects a random sample of 100 readings at different times and dates within the two-mile range around the plant. The findings are that the sample average concentration is 60 ppm and the sample standard deviation is 20 ppm. Is there evidence to conclude that the plant in question is violating the law?

The data obtained for this study are as follows:

n=100;[pic]= 60;s = 20;For α = 0.01, critical value of z is 2.33

H0: μ ≤ 55[one-tail test: right/upper tail]
H1: μ >55

The computed value of Z is

[pic]

Since the test statistic falls in the rejection region, H0 is rejected, and we may conclude that the average concentration of vinyl chloride is more than 55 ppm.

Example 3

A certain kind of packaged food bears the following statement on the package: “Average net weight 12 oz.” Suppose that a consumer group has been receiving complaints from users of the product who believe that they are getting smaller quantities than the manufacturer states on the package. The consumer group wants, therefore, to test the hypothesis that the average net weight of the product in question is 12 oz. versus the alternative that the packages are, on average, underfilled. A random sample of 144 packages of the food product is collected, and it is found that the average net weight in the sample is 11.8 oz. and the sample standard deviation is 6 oz. Given these findings, is there evidence the manufacturer is underfilling the packages?

The data obtained for this study are as follows:

n=144;[pic]= 11.8;s = 6;For α = 0.05, the critical value of z is -1.645

H0: μ ≥ 12[one-tail test: left/lower tail]
H1: μ < 12

The computed value of Z is

[pic]

Since the test statistic falls in the nonrejection region, H0 is not rejected, and we may not conclude that the manufacturer is underfilling packages on average. 2. Test Statistic for the Population Mean ((): Small Sample Test

When the population is normal, the population standard deviation, σ, is unknown and the sample size is small, the hypothesis test is based on the t distribution, with (n-1) degrees of freedom, rather...
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