What is a Test of Significance?

Statistical hypotheses – statements about population parameters

 ExamplesMean weight of adult males is greater than 160Proportion of students with a 4.0 GPA is less

than .01

In statistics, we test one hypothesis against another The hypothesis that we want to prove is called the

alternative hypothesis,  Another hypothesis is formed that contradicts .

This hypothesis is called the null hypothesis,

After taking the sample, we must either: Reject and believe , or Fail to reject because there was not sufficient evidence to reject it (meaning there is not sufficient evidence to prove )

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aH

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Types of errors

The probability with which we are willing to risk a type I error is called the level of significance of a test and is denoted

 The probability of making a type II error is

denoted

Fail to reject Reject

is true Correct Type I error

is false Type II error Correct

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The quantity is known as the power of a test. It represents the probability of rejecting when in fact it is false

Decreasing increases Sample size is the only way to control both

types of error

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Test Statistic – the statistic we compute to make the decision (sampling distribution of the test statistic must be known)

 The p-value of a hypothesis test is the smallest

value of such that would have been rejected

 If , reject If , fail to reject

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value-p value-p

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Steps of a hypothesis test1) State and 2) Calculate the test statistic3) Identify the p-value4) Make decision and interpret results

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ExampleThe current treatment for a type of cancer

produces remission 20% of the time. An investigator wishes to prove that a new method is better. Suppose 26 of 100 patients go into remission using the new method.

There is not sufficient evidence to conclude the new method is better.

05.

ExampleDo less than 50% of people prefer Murray’s

Vanilla Wafer’s when compared to other brands? Suppose that in a taste test 42 of the 250 choose Murray’s.

Conclude with 95% confidence that less than 50% of people prefer Murray’s Vanilla Wafer’s when compared to other brands.

05.

Inference about a Population MeanRemember

is the standard deviation of the sampling distribution which is referred to as the standard error

has approximately a standard normal distribution

Therefore,and the confidence interval is

x

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n

xZ

)(n

ZE

Ex

ExampleA sample of 100 visa accounts were studied for the

amount of unpaid balance. andConstruct a 95% confidence interval

We are 95% confident the mean unpaid balance of visa accounts is between $619.13 and $670.87.

Construct a 99% confidence intervalWe are 99% confident the mean unpaid balance of visa accounts is between $611.00 and $679.00.

Notice that as we increase the confidence level the interval gets wider

645$x 132$

ExampleA random sample of 500 apples yields

Assume

Find a 95% confidence interval

We are 95% confident the population mean weight of apples is between 9.104 and 9.296 oz.

oz. 2.9xoz. 1.1

ExampleA consumer protection agency wants to prove

that packages of Post Grape Nuts average less than 24 oz.

Conclude with 95% confidence that packages of Post Grape Nuts has a mean less than 24 oz.

05.100n

94.23x13.

ExampleIt is desired to show the mean weight of a

metal component is greater than 4.5 oz.

There is not sufficient evidence to prove that the mean weight is greater than 4.5 oz.

10

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