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What is a Test of Significance?
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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
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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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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.
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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.
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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
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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
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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
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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.
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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.
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