Lecture 27

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Hypothesis Testing for Population Proportion

Example

Poll was conducted just before election day for the key senate race in New York between Senator D’Amato and Rep Schumer. The analsyst wants to use the poll results to test the hypothesis that the incumbent, Senator D’Amato will win the election. The winner needs a simple majority of the voters.

What are the hypotheses for this test? (We are concerned here with the proportion for Schumer.)

Ho: pop. proportion is less than or equal to po

or

Ho: pop. proportion is greater than or equal to po

What is po for this test?

In this case we are using sample proportion (p-hat) to test for the population proportion (p)

 

Hypothesis Testing for Population Proportion

The procedure for hypothesis testing of population proportions is essentially the same as for hypothesis testing for population means. The disfference is in the sampling distribution of p-hat.

Sampling Distribution of p-hat

mean of sampling distribution = pop proportions

stan deviation of sampling distribution = sqrt (pq/n)

sampling distribution is normal if n is large enough

Sample size is considered large if the interval

p-hat + or - 3 s{p-hat}

does not include 0 or 1

 

Hypothesis Testing for Population Proportion

The results of the poll of 500 New Yorker showed that Rep Schumer had a level of support of 53%.

Test the following hypotheses at 95% confidence

Ho: pop. proportion is less than or equal to 0.50

 

 

(You need to test whether or not the sample size is large enough)

State the conclusions in Statistical terms and in plain english.

Would the conclusions change if we use a 90% level of confidence.

What is the p-value for this test?

Why is it better to report the results in terms of the p-value?

 

Lecture 28

 

Lecture 29

 

 

Hypothesis Testing for Population Variance

Example

The engineer in charge of pavement compaction on a large interstate project wants to ensure that the pavement not only meets minimum level of compact but that the compaction is also uniform over the full surface of the mat.

The degree of uniformity is assessed by looking at the variance of the compaction density. The regulating agent requires that the population variance is no more than 5 pounds per cubic feet.

The engineer develop a monitoring system in which density is measured by nuclear density gauge at intervals of 20 feet along the pavement. This sample result is used to assess whether or not the compaction meets the criteria for uniformity.

 

This is an example were hypothesis testing for population variance is required

The hypothesis is of the form

Ho: pop. variance is greater than or equal to the standard

 

 

Sample variance is used to conduct this hypothesis test

The test is based on the following quantity: (n-1)s2/s2

This sampling distribution of this quantity has a Chi-squared (C2) distribution when the sample is taken from a population which is normally distributed

 

Hypothesis Testing for Population Variance

Results for the sample

Sample mean = 148.6 pcf

Sample standard deviation = 2.05 pcf

Number of measurement points = 24

Use this result to test whether or not the pavement meet the uniformity criteria (population variance should be no more than 5 pounds per cubic feet). The alpha-level is 5 %.

Ho: pop. variance is greater than or equal to the standard

 

 

Test Statistic = C2 = (n-1)s2/s2

(the pop. variance is assumed to be s2o)

C2 = 23*2.052/5=19.33

Lower Tail rejection - we need C2( n-1=23: 1-alpha)

Table VII C2(23: 0.95) = 10.20

Test Statistic is in the acceptance region: we cannot reject Ho.

 

Chi-square Distribution

Note

The Chi-square distribution is NOT symmetrical.

The Chi-square distribution goes from 0 to infinity.

 

 

Test the following hypothesis at a 95 % level of confidence

Ho: pop. variance is equal to the standard

 

 

Results for the sample

Sample mean = 148.6 pcf

Sample standard deviation = 2.05 pcf

Number of measurement points = 24