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T09-00 1 Population Hypothesis Tests

Purpose Allows the analyst to analyze the results of hypothesis testing for known large sample means, small or unknown sample means, and proportion situations based on an a significance level. The hypothesis tests include the 3 alternative options of "not equal", "greater than", and

"less than" with appropriate conclusion, p-value, test statistic, and critical value(s).

Inputs level Sample size Hypothesized value Hypothesis alternative Sample statistics (mean, standard deviation, proportion)

Outputs Test Statistic Critical Value(s)p-valueHypothesis test conclusion

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Critical Values

Reject Ho

- CV

-CV = Z/t such that P(<Z/t) =

Reject Ho

CV

CV = Z/t such that P(>Z/t) =

-CV = Z/t such that P(<Z/t) = CV = Z/t such that P(>Z/t) =

Reject Ho

- CV

Reject Ho

CV

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p-value (Right Tail Test)

One tail to the rightHo: Ha:

For one tail tests to the right (>) the p-value is the probability greater than the test statistic

Test statistic

p-value

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p-value (Left Tail Test)

One tail to the leftHo: Ha:

For one tail tests to the left (<) the p-value is the probability greater than the test statistic

p-value

- Test statistic

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p-value (Two Tail Test)

Two tail Ho: Ha:

For two tail tests to the left () the p-value is the probability greater than the right test statistic plus the probability less than the left test statistic

1/2 p-value 1/2 p-value

Test statistic

- Test statistic

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Test Statistic = Z =X -

n

o

Methodology Assumptions

Large samples (n >= 30)

Population Variance is known

Hypothesis Test for Means (Known Variation)

Left Tail: Z

Right Tail: - Z

Two Tail: - Z and Z

/ /2 2

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The Federal Trade Commission (FTC) periodically conducts studies designed to test the claims manufacturers make about their products. For example the label on a large can of Hilltop Coffee states that the can contains at least 3 pounds of coffee.

A random sample of 36 cans of coffee were tested yielding a sample mean of 2.92 lbs per can. Previous studies have shown the standard deviation in the lbs/can to be .18.

Is Hilltop Coffee making a valid claim? Test their claim for a 99% confidence level.

Hypothesis Test Means (Known Variation) - Example

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One tail to the leftHo: 3

Ha: 3

Test statistic = -2.667

Facts:CV = -2.326 test statistic = -2.667 p-value = .004 .01 null hypothesis is false alternative is true

Reject Ho

-2.326 Business Conclusion:

The statistical evidence shows that Hilltop Coffee is is not living up to their claim (at least 3 pounds of coffee in each can).

Hypothesis Test Means (Known Variation) - Example

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Input the and the sample mean, standard deviation, sample size and hypothesized value

Select the alternative hypothesis

The test statistic, critical values, p-value and decision are automatically calculated

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Test Statistic = t =X -

s

n

o

Hypothesis Test for Means (Unknown Variation)

Small sample (n < 30)

Population Variance is unknown Methodology Assumptions

Left Tail: t

Right Tail: - t

Two Tail: - t and t

,

,

/ , / ,

df

df

df df2 2

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The manufacturing manager of the Hiney Winery is responsible for filling the Tiny Hiney wine bottles with 16 oz of wine. If the wine bottles are too full the Hiney Winery looses money and if they are do not contain 16 ounces their customers get upset.

A random sample of 8 bottles are sampled yielding the following fill in ounces.

16.02, 16.22, 15.82, 15.92, 16.22, 16.32, 16.12, and 15.92.

Using a level of significance of .05 determine whether the Heiney Winery is filling their bottles with an average of 16 oz of wine.

16 oz

Tiny Hiney

Hypothesis Test Means (Unknown) - Example

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Two tail Ho: = 16Ha: 16

Test statistic = 1.1186

Reject Ho

- 2.3646

Reject Ho

2.3646

Facts:CV = -2.3646 & +2.3646test statistic = 1.1186 p-value = .3002.05 null hypothesis is true alternative is false

Business Conclusion:

The statistical evidence shows that the Hiney Winery is putting 16 ounces of wine in their Tiny Hiney bottles.

Hypothesis Test Means (Unknown) - Example

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Input the and the sample mean, standard deviation, sample size and hypothesized value

Select the alternative hypothesis

The test statistic, critical values, p-value and decision are automatically calculated

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Test Statistic = Z =p -

n

o

o

( )

1

Hypothesis Test for Proportions (Large)

Methodology Assumptions

Sample size is sufficiently large

Sufficiently Large (np 5 and n(1- p) 5)

Left Tail: Z

Right Tail: - Z

Two Tail: - Z and Z

/ /2 2

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Over the past few weeks the Bushwood Golf Course 20% of the players have been women. In an effort to attract more women players the head pro has developed a special “drive time advertising” promotion to attract women golfers to Bushwood.

After a week of running the promotion, a random sample of 400 golfers showed that 300 of them were men and 100 were women.

Use a level of significance of .05 to determine if promotion has had an impact on the the number of women playing at Bushwood.

Hypothesis Test for Proportions (Large) - Example

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Test statistic = 2.50

One tail to the rightHo: .20Ha: .20

Reject Ho

1.6449

Facts:CV = 1.6449 test statistic = 2.500 p-value = ..0062 .05

null hypothesis is false alternative is true

Business Conclusion:

The statistical evidence shows the “drive time” advertising promotion has been effective in attracting more women to play Bushwood.

Hypothesis Test for Proportions (Large) - Example

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Input the and the sample proportion, sample size and hypothesized value

Select the alternative hypothesis

The test statistic, critical values, p-value and decision are automatically calculated