Statistics – Ch 9.1 Notes Name: ___________________ Significance Tests: The Basics

1. State the null and alternative hypotheses for a significance test about a population parameter. State the appropriate hypotheses for Mr. McGee’s free throw shooting claim.

In any significance test, the null hypothesis has the form

H0 : parameter = value

The alternative hypothesis has one of the forms Ha : parameter < value Ha : parameter > value Ha : parameter ≠ value

One-Sided vs Two-Sided Tests

Stating Hypotheses:

1. The hypotheses should express the hopes or suspicions we have before we see the data. It is cheating to look at the data first and then frame hypotheses to fit what the data show.

2. Hypotheses always refer to a population, not to a sample. Be sure to state 𝐻0 and 𝐻𝑎 in terms of population parameters

3. It is never correct to write a hypothesis about a sample statistic, such as �̂� = 0.64 𝑜𝑟 �̅� = 85

Practice Problem: At the Hawaii Pineapple Company, managers are interested in the size of the pineapples grown in the company’s fields. Last year, the mean weight of the pineapples harvested from one large field was 31 ounces. A different irrigation system was installed in this field after the growing season. Managers wonder how this change will affect the mean weight of pineapples grown in the field this year. PROBLEM: State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

The alternative hypothesis is one-sided if it states that a parameter is larger than the null hypothesis value or if it states that the parameter is smaller than the null value.

It is two-sided if it states that the parameter is different from the null hypothesis value (it could be either larger or smaller).

Keynull 0HoiP 0.80

alternative0Ha Pao 80

onesidedT Twosided

Ho µ 31oz

Ha M 31oz

Statistics – Ch 9.1 Notes Name: ___________________ Significance Tests: The Basics

2. Interpret a P-value in context. Calculate and interpret the p-value from Mr. McGee’s free throw shooting test.

The null hypothesis 𝐻0 states the claim that we are seeking evidence against. The probability that measures the strength of the evidence against a null hypothesis is called a P-value.

• ________________ P-values are evidence against 𝐻0 because they say that the observed result is unlikely to occur when 𝐻0 is true.

• ________________P-values fail to give convincing evidence against 𝐻0 because they say that the observed result is likely to occur by chance when 𝐻0 is true.

3. Determine if the results of a study are statistically significant and draw an appropriate conclusion

using a significance level.

Statistical Significance: when the results of a test are so extreme that they could not have occurred by chance alone.

Conclusions of a Test: Note: A fail-to-reject H0 decision in a significance test doesn’t mean that H0 is true. For that reason, you should never “accept H0” or use language implying that you believe H0 is true.

In a nutshell, our conclusion in a significance test comes down to

P-value small → _________________________ → convincing evidence for Ha

P-value large → _________________________ → not convincing evidence for Ha

The probability computed, assuming H0 is true, that the statistic would take a value as extreme as or more extreme than the one actually observed is called the P-value of the test.

AssumingMrMcGee is an 80 free throw shooter thereis a probability of getting a sampleproportion of 0.64or less bychancealone

Small

Large

reject Hofail torejectHo

Statistics – Ch 9.1 Notes Name: ___________________ Significance Tests: The Basics

Practice Problem: Calcium is a vital nutrient for healthy bones and teeth. The National Institutes of Health (NIH) recommends a calcium intake of 1300 mg per day for teenagers. The NIH is concerned that teenagers aren’t getting enough calcium. Is this true? Researchers want to perform a test of

where μ is the true mean daily calcium intake in the population of teenagers. They ask a random sample of 20 teens to record their food and drink consumption for 1 day. The researchers then compute the calcium intake for each student. Data analysis reveals that �̅� = 1198𝑚𝑔 𝑎𝑛𝑑 𝑠𝑥 = 411 𝑚𝑔. After checking that conditions were met, researchers performed a significance test and obtained a P-value of 0.0729.

a) Explain what it would mean for the null hypothesis to be true in this setting.

b) Interpret the P-value in context.

c) What conclusion would you make at α = 0.10 significance level? Justify your answer.

d) What conclusion would you make at α = 0.05 significance level? Justify your answer.

HoM 1300 thedailycalciumintakeinthepopulation ofteenagersis 1300mg Iftruethenteenagersare gettingenoughcalciumonaverage

Pvalue0.0729 assumingthemeandailycalciumintake ofteenagers is Boothereis a0.0729probability of getting a sample mean of 1198mg or lessbychancealone

RejectHo because our pvanlue0.0229 2 0.10we haveconvincingevidence that teenagers are notgetting1300mg of calcium perday onaverage

fail to rejectHo because our pvalue 0.0729 2 2 0.05we do not haveconvincingevidence that the meanintake of calciumfor teenagers is less than 1300mgperday

Statistics – Ch 9.1 Notes Name: ___________________ Significance Tests: The Basics

4. Interpret a Type I and a Type II error in context and give a consequence of each. Type I and Type II Errors When we draw a conclusion from a significance test, we hope our conclusion will be correct. But sometimes it will be wrong. There are two types of mistakes we can make.

Significance and Type I Error

Practice Problem: A potato chip producer and its main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. The producer will then perform a significance test using the hypotheses

where P is the actual proportion of potatoes with blemishes in a given truckload. PROBLEM: Describe a Type I and a Type II error in this setting, and explain the consequences of each.

If we reject H0 when H

0 is true, we have committed a ________________________ error.

If we fail to reject H0 when H

a is true, we have committed a _____________________ error.

The significance level α of any fixed-level test is the probability of a Type I error.

That is, α is the probability that the test will reject the null hypothesis H0 when H

0 is actually true.

Consider the consequences of a Type I error before choosing a significance level.

Type IType I

T.yfe.LE.EE WerejectHowhenHo is true Theproducerfindsconvincingevidencethattheproportion of blemishes is greaterthan0.8whentheactual proportion is 0.8or less Consequenceswould besendingan acceptable truckloadofpotatoesback resultinginlossofrevenue

Iffe ErE I failtorejectHowhenHaistrueTheproducerdoesnotfindconvincingevidencethatmorethan 8 ofthepotatoeshaveblemisheswhenthatactually is thecase Consequenceswouldbeusingthetruckload ofpotatoes tomakechipsMorechipswillbemadewithblemishedpotatoes whichmayupset

customers