Fundamentals of Hypothesis Testing

Chapter 10Created by Laura Ralston

Outline

• 10.1 Fundamentals of Hypothesis Testing• 10.2 Hypothesis Testing for Means (Small

Samples)• 10.3 Hypothesis Testing for Means (Large

Samples)• 10.4 Hypothesis Testing for Population

Proportions ?• 10.5 Types of Errors• 10.6 Hypothesis Testing for Population Variance• 10.7 Chi-Square Test for Goodness of Fit• 10.8 Chi-Square Test for Association

Introduction

• Anyone can make a claim about a population parameter– “Four out of five dentists prefer StarBrite

toothpaste”– “Fewer than 1% of dogs attack people unprovoked”– “In any given one-year period, ….about 18.8 million

American adults suffers from a depressive illness” –Reader’s Digest

– “For cars, this Corporate Average Fuel Economy has been 27.5 miles per gallon since 1990….” USA Today

• People make decisions everyday based on such claims. – Some decisions are insignificant (which toothpaste

to buy)– Others are very important!• Whether to use a new prescription drug that has just

come onto the market• Purchase a car based on reported safety ratings• Are public safety campaigns working?• Which candidate is likely to win an election?

Hypothesis Testing

• One statistical process that sets a uniform standard for evaluating claims about a population parameter

• Foundation for hypothesis testing is the RARE EVENT RULE: if, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct (Gender selection example from Triola)

Step 1: State the hypotheses in mathematical terms

Null Hypothesis, H0

• Claim about a population parameter

• Always contains “equals”

Alternative Hypothesis, H1 or Ha

• Claim about a population parameter

• Opposite of the null hypothesis

Indicate which hypothesis is stated in the given problem (o.c.). This information is needed later.

Common Phrases>Is greater thanIs aboveIs higher thanIs longer thanIs bigger thanIs increased

<Is less thanIs belowIs lower thanIs shorter thanIs smaller thanIs decreased or reduced from

>Is greater than or equal toIs at leastIs not less than

<Is less than or equal toIs at mostIs not more than

=Is equal toIs exactly the same asHas not changed from Is the same as

≠Is not equal to Is different from Has changed from Is not the same as

Step 2: Determine the critical value(s)

• Based on the level of significance, a, which will be given in the problem and using normal distribution

• Critical value(s) separates “guilty” and “not guilty” • Use calculator: invNorm command• “Type” of test is determined by Ha which will

determine the critical region(s) and ultimately the critical value(s) – Right-Tailed test if > or > (picture)– Left-Tailed test if < or < (picture)– Two-tailed test if ≠ (picture, a/2)

Step 3: Calculate test statistic

• Test statistic will vary depending on the parameter and sample size (z, t, or chi-square)

• Used to make a decision about the null hypothesis, “evidence”

• Calculated using sample data (given in problem)

Step 4: Make a decision

• Variety of methods (traditional, p-value, confidence intervals)

• Two possibilities– Reject the Null Hypothesis -assumption is not valid

based on this set of sample data – Fail to reject the Null Hypothesis (never “accept”)-

assumption is valid based on this set of sample data

Step 5: State final conclusion in non-technical terms

• Conclusion should be well-worded, reference the original claim, and free of mathematical symbols.

• See flow chart on next slide to help determine wording

Does the original claim contain the condition of

equality?

Do you reject H0 ?

“There is sufficient evidence to warrant rejection of the

claim that …..(original claim)

“There is not sufficient evidence to warrant

rejection of the claim that …(original claim).”

Do you reject H0?

“The sample data support the claim that ….(original

claim).”

“There is not sufficient sample evidence to support

the claim that ….(original claim)”

YES (original claimcontains equality)

NO (original claim doesnot contain equality)

Yes

No

Yes

No

Example 1

• In a sample of 27 blue M&Ms with a mean weight of 0.8560 g. Assume that s is known to be 0.0565 g. Consider a hypothesis test that uses a 0.05 significance level to test the claim that the mean weight of all M&Ms is equal to 0.8535 g (the weight necessary so that bags of M&Ms have the weight printed on the packages).

Example 2

• In a sample of 106 body temperatures with a mean of 98.20°F. Assume that s is known to be 0.62°F. Consider a hypothesis test that uses a 0.05 significance level to test the claim that the mean body temperature of the population is less than 98.6°F.

Example 3

• The average production of peanuts in the state of Virginia is 3000 pounds per acre. A new plant food has been developed and is tested on 60 individual plots of land. The mean yield with the new plant food is 3120 pounds of peanuts per acre with a standard deviation of 578 pounds. At a 0.05 significance level, can one conclude that the average production has increased?

Example 4

• A researcher claims that the yearly consumption of soft drinks per person is 52 gallons. In a sample of 50 randomly selected people, the mean of the yearly consumption was 56.3 gallons. The standard deviation of the sample was 3.5 gallons. At a 0.01 significance level, is the researcher’s claim valid?

Assignment

• Use traditional method ---modeled in above examples

• Page 498 #25-30