HYPOTHESIS TEST FOR ONE POPULATION PROPORTION, Study notes of Probability and Statistics

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HYPOTHESIS TEST FOR

ONE POPULATION

PROPORTION – STEP 3

Unit 4A - Statistical Inference Part 1

Now let’s look at STEP 3 for the z‐test for the population proportion (p). Here • Use we our test statistic to

• Determine the p‐value of our test The the nullp‐value hypothesis is the probability is true. of observing data as or more extreme than our data assuming

This ourselves test for using one normal proportion probabilities. will be the For only all situationremaining where tests inwe this will course, find the we p ‐willvalue rely on software to obtain the p‐values for us. We are using this situation to help us understand p‐values so that we will be more able to correctly use and understand the results we obtain from software for future tests.

Both the test statistic and the p‐value are measures of the evidence against Ho. Although researchers conclusions use the p (^) ‐canvalue be along drawn with directly the 5% from significance test statistic level values, to determine in practice whether most or not to reject the null hypothesis. For TEST The STATISTICS LARGER the test statistic, The further p‐hat is from p 0 And the MORE EVIDENCE AGAINST Ho But for P The ‐VALUES SMALLER it is (^) thethe popposite.‐value, The further p‐hat is from p 0 And the MORE EVIDENCE AGAINST Ho One interpretable benefit of and using easy p‐ valuesto work is with that asthe z (^) ‐valuesscores ofwhich test statisticsfollow a normal may not distribution. always be as P‐values have the same interpretation for every test and therefore we can develop a unified approach to STEPS 3 and 4 for all tests regardless of the results from STEP 2.

P-VALUES

The P-VALUE is the

• probability of
• observing a test statistic as extreme as that observed (or even more extreme)
• assuming that the null hypothesis is true

By “extreme” we mean extreme in the

direction(s) of the alternative hypothesis.

Null Distribution

 Test Statistic:

 If our conditions are satisfied:

• Random Sample
• Sample Size Large Enough

 Then, assuming Ho is true, this test statistic is distributed as a STANDARD NORMAL DISTRIBUTION

 We use this distribution to find our p-values!

Recall our test statistic equation which takes the form of a z‐score for p‐hat. From random our sample work on and sampling a large enoughdistributions, sample if (^) relativeour two toconditions the hypothesized our satisfied proportion, – having a

then under the assumption that the null hypothesis is true our test statistic is distributed as a with standard already normal using (^) zdistribution‐scores, to calculateand we can p‐values. use this distribution, which we have worked

Notice intervals that but we there went we through used the similar normal arguments distribution in our to discussionfind cutoffs about for the confidence appropriate confidence level. Here we will use the normal distribution to find probabilities. Maybe you can see now why we needed to learn about normal distributions and be able to work confidence with them, intervals we neededto find the it here cutoffs to be required. able to discuss p‐values and we needed it for

If the alternative hypothesis is Ha: p < p 0 (less than) , then “extreme” means small or less than , and the p‐value is:

• The probability of observing a test statistic as small as that observed or smaller if the null hypothesis is true. We case can z. This write notation this in notation is sometimes as P(Z confusing ≤ our test for statistic) students. which Here here the capital we denote Z represents by the lower the DISTRIBUTION theoretically – the random variable. And the lower case z represents the value of our test statistic. Hopefully when we look at our examples this will be clear. You do not need to use this notation but you should start with writing the probability you are calculating with your test statistic instead of lower case z. So, if you have a test statistic of z = ‐2.5 then we would write P(Z ≤ ‐2.5)

Ha is “Greater Than”

Example: P(Z ≥ 1.5)

Ha is “Not Equal To”

Examples: Z = -2.5  2P(Z ≥ 2.5) Z = 1.5  2P(Z ≥ 1.5)

If the alternative is Ha: p ≠ p 0 (different from) , then “extreme” means extreme in either direction value therefore either (^) is: small or large (i.e., large in magnitude) or just different from , and the p‐

• The larger probability if the null of hypothesis observing is a true.test statistic as large in magnitude as that observed or

Here are two examples: If • z (^) The= ‐2.5: p‐value = probability of observing a test statistic as small as ‐2.5 or smaller or as large as 2.5 or larger. If z = 1.5:

• The as ‐1.5 p‐value or smaller. = probability of observing a test statistic as large as 1.5 or larger, or as small

The statistic easiest – in way other to wordsfind the – remove p‐value (^) anyin this negative case is signs to take and the then absolute find the value right of‐tailed our test probability: P(Z ≥ the absolute value of our test statistic) and then double this value. Our test generalstatistic). formula for the p‐value in this case is 2 times P(Z ≥ the absolute value of our

No and matter lower tails.how youYou findcan dothis this p‐value, by you need to make sure that you obtain both the upper

• Finding as they bothare the probabilities same value and so itadding is easier them to together – you will likely tire of this method
• Pick either tail – but only one of them and then find the probability, and finally double the result. Now let’s return to our examples.

Has the proportion of defective products been reduced as a result of the repair? The • The p‐value probability in this ofcase observing is: a test statistic as small as ‐ 2 or smaller, assuming that Ho is true. OR • The (recalling probability what of the observing test statistic a sample actually proportion means (^) in that this is (^) case), 2 standard errors or more

OR,^ below more^ the specifically,^ null^ value^ (p^0 =^ 0.20),^ assuming^ that p^0 is^ the^ true^ population^ proportion.

• The probability of observing a sample proportion of 0.16 or lower in a random sample of size 400, when the true population proportion is p 0 =0. The p‐value is found as shown in the normal curve here.

Example 1: Defective Products

P-Value = P(Z ≤ -2.0) = 0.

Example 2: Marijuana Use

 Ho: p = 0.

 Ha: p > 0.

P-Value = P(Z ≥ 0.91)

Is the proportion of marijuana users in the college higher than the national figure? The • The p‐value probability in this ofcase observing is: a test statistic as large as 0.91 or larger, assuming that Ho is true. OR • The (recalling probability what of the observing test statistic a sample actually proportion means (^) in that this is (^) case), 0.91 standard errors or more

OR,^ above more^ the specifically,^ null^ value^ (p^0 =^ 0.157),^ assuming^ that p^0 is^ the^ true^ population^ proportion.

• The probability of observing a sample proportion of 0.19 or higher in a random sample of size 100, when the true population proportion is p 0 =0. The p‐value is found as shown in the normal curve here.