Section 10.2

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Section 10.2

Hypothesis Testing for Population Means (s Known)

With valuable content added by D.R.S., University of Cordele






Two ways to reach a conclusion

1. Use the rejection region to draw a conclusion. 2. Use the p-value to draw a conclusion.

There are two ways to get the answer and we learn BOTH.• The “Critical Value” method, comparing your z Test Statistic

to the z Critical Value, which was determined by the chosen α Level of Significance.

• The “p-Value method”, in which your z Test Statistic leads to a “p-Value”, an area under the normal curve, which is compared to the chosen α Level of Significance.







Test Statistic for a Hypothesis Test for a Population Mean (s Known) When the population standard deviation is known, and

either the sample size is at least 30 or the population distribution is approximately normal, THEN COMPUTE: the test statistic for a

hypothesis test for a population mean is given by

xz

n

s

is _________________________________

is ___________________ according to ____

is _________________________________

is _________________________________







THEN COMPUTE: the test statistic for a hypothesis test for a population mean is given by

xz

n

s

My sample mean is how far away from the H0 Null Hypothesis mean?

Gauged by the standard deviation for a sample mean as defined in the ___________ _________ Theorem.

We will use this Test Statistic to decide between

___________ing H0 or ___________ing to _________ H0.






Rejection Regions

Value Left-tailed testValue Right-tailed testValue Two-tailed test

Alternative Hypothesis, Type of Hypothesis TestaH

The “rejection region” is the one-tail area of size α.

The “rejection region” consists of two tails, each of them of size α/2, left and right.

And the “Fail to Reject H0” region is the big area of size 1 – α.






A Left-Tailed Hypothesis Test: Ha < μ

The critical z value is the negative z value which separates the left tail of area α.

DECISION RULE: Reject the null hypothesis, H0, if the test statistic calculated from the sample data falls within the rejection region.






A Right-Tailed Hypothesis Test: Ha > μ

The critical z value is the positive z value which separates the right tail of area α.







A Two-Tailed Hypothesis Test: Ha ≠ μ

The critical z values are the +/- z values which separate the two tails, area α/2 each.


Commonly-occurring critical values

Area in tail(s),

α, alpha(and c too)

One-Tailed Test

Two-Tailed Test

0.10 (0.90) 1.28 ±1.6450.05 (0.95) 1.645 ±1.960.02 (0.98) 2.05 ±2.330.01 (0.99) 2.33 ±2.575

• These values for the α Level Of Significance (or c Level Of Confidence) are typical. You should already know how to find them using tables and using invNorm(). It is convenient to have this special table for reference.






Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right Tailed, ‑ s Known)

The state education department is considering introducing new initiatives to boost the reading levels of fourth graders. The mean reading level of fourth graders in the state over the last 5 years was a Lexile reader measure of 800 L. (A Lexile reader measure is a measure of the complexity of the language that a reader is able to comprehend.) The developers of a new program claim that their techniques will raise the mean reading level of fourth graders by more than 50 L.






Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right Tailed, ‑ s Known) (cont.)

To assess the impact of their initiative, the developers were given permission to implement their ideas in the classrooms. At the end of the pilot study, a simple random sample of 1000 fourth graders had a mean reading level of 856 L. It is assumed that the population standard deviation is 98 L. Using a 0.05 level of significance, should the findings of the study convince the education department of the validity of the developers’ claim?







Solution Step 1: State the null and alternative hypotheses.

The developers want to show that their classroom techniques will raise the fourth graders’ mean reading level to more than 850 L.

H0: ____________ Ha:______________







Step 2: Determine which distribution to use for the test statistic, and state the level of significance.

Since the population parameter of interest is _______ and s is { known, unknown }, the sample is a simple random sample, and the sample size is ≥ ______. Thus, we will use a _______ distribution, and { z, t } . And someone decided that in this experiment, the level of significance is a = _________







.

DRAW A PICTURE!Label the critical z value.Later, put your test statistic at theappropriate place on the z axis.







Step 3: Gather data and calculate the necessary sample statistics.

(They’ve done all the hard work for us.)“At the end of the pilot study, a simple random sample of 1000 fourth graders had a mean reading level of 856 L. The population standard deviation is 98 L.”

Observation: The 856 { is, is not } more than the ______ in the null hypothesis, but is it strongly enough higher to be s____________ally s___________ ?







Calculate your test statistic. Write it in on the appropriate place on the z-axis in your normal curve picture. (a couple of slides ago)

𝑧=𝑥−𝜇

( 𝜎√𝑛 )







Step 4: Draw a conclusion and interpret the decision. Remember that we determine the type of test based on the alternative hypothesis. In this case, the alternative hypothesis contains “>,” which indicates that this is a right tailed test. ‑

Recall that we picked α = _______The critical value, the z value that has this areato its right, is __________. (And update your picture.)







The z-value of 1.94 falls { inside or outside ? } the rejection region. So the conclusion is to { Reject? or Fail To Reject? } the Null Hypothesis.

Conclusion: Thus the evidence collected suggests that the education department can be _____% sure of the validity of the developers’ claim that the mean Lexile reader measure of fourth graders will increase by more than 50 points.






p-Values

p-valueA p-value is the probability of obtaining a sample statistic as extreme or more extreme than the one observed in the data, when the null hypothesis, H0, is assumed to be true.

We’ll give examples of how to find a p-value by hand.But usually you’ll get the p-value for free from the TI-84’s ZTest or TTest feature.






Example: Calculating the p-Value for a z-Test Statistic for a _______-Tailed Test

. 0: 0.15: 0.15a

HH

The p-value is the area to the { left right } of z= _______

The p-value is ______________

Calculate the p-value for a hypothesis test with the hypothesesshown at right.. Assume that data have been collected and the test statistic was calculated to be z = 1.34.






Example: Calculating the p-Value for a z-Test Statistic for a _______-Tailed Test

.

The p-value is the area to the { left right } of z= _______

The p-value is ______________

Calculate the p-value for a hypothesis test with the hypothesesshown at right.. Assume that data have been collected and the test statistic was calculated to be z = 2.78.

0: 0.43: 0.43a

HH






Example Calculating the p-Value for a z-Test Statistic for a ______-Tailed Test

.

We find the area to the { left right } of z= _______

The p-value is ______________ times _____ = __________.

Calculate the p-value for a hypothesis test with the hypothesesshown at right.. Assume that data have been collected and the test statistic was calculated to be z = – 2.15.

0: 0.78: 0.78a

HH






How to make a conclusion using the p-Value

Conclusions Using p-Values • If p-value ≤ a, then reject the null hypothesis. (because your small p-value, that small area that’s more extreme than your extreme result, is inside the larger rejection area α)• If p-value > a, then fail to reject the null hypothesis.

Z-Test for the Reading Lexile Scores problem

STAT, TESTS, 1:Z-Test. Since they gave us summary statistics, we pick “Stats” (not “Data”, which is for a problem where we have the individuals’ scores).μ0 comes from H0

σ is the “known” stdev.bar-x from our samplen is our sample sizeGive the Ha inequalityHighlight Calculate and press ENTER.

Z-Test for this problem.

STAT, TESTS, 1:Z-Test. Here are the results.It reminds you of what you told it was the alternative hypothesis. Have a look and make sure it’s what you wanted.It tells you the Test Statistic.It tells you the p-Value.It reminds you of the sampledata, the sample mean andsample size you told it.

Z-Test for this problem.

So for the reading example, the p-value was ________ and the level of significance was ________. Therefore we { reject, fail to reject } the null hypothesis.






Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, s Known)

A researcher claims that the mean age of women in California at the time of a first marriage is higher than 26.5 years. Surveying a simple random sample of 213 newlywed women in California, the researcher found a mean age of 27.0 years. Assuming that the population standard deviation is 2.3 years and using a 95% level of confidence, determine if there is sufficient evidence to support the researcher’s claim.

Example 10.16: TI-84 Z-Test solution

• Null Hypothesis: ________ Alternative: ________What is the α Level Of Significance? α = _______What inputs do you give to the TI-84 Z-Test?• _______, ________• _______, ________• : ≠ or < or >The TI-84 Z-Test output gives us a p-value of ________.Therefore we { reject or fail to reject } H0.Draw a bell curve diagram of the story:






Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, s Known) (cont.)

Draw a bell curve diagram for this problem.

Could you calculate the z test statistic and find the critical z value if the problem asked for those, too?






Examples 10.16 and 10.17 complete solutions by hand with all the details

Complete by-hand detailed solution for Example 10.16:and theComplete by-hand detailed solution for Example 10.17, the next problem1) 2205.drscompany.com2) click on Slides3) The links are there, immediately following the link

for these 10.2 slides.






Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, s Known)

A recent study showed that the mean number of children for women in Europe is 1.5. A global watch group claims that German women have a mean fertility rate that is different from the mean for all of Europe. To test its claim, the group surveyed a simple random sample of 128 German women and found that they had a mean fertility rate of 1.4 children. The population standard deviation is assumed to be 0.8. Is there sufficient evidence to support the claim made by the global watch group at the 90% level of confidence?

Example 10.17: TI-84 Z-Test solution

• Null Hypothesis: ________ Alternative: ________What is the α Level Of Significance? α = _______What inputs do you give to the TI-84 Z-Test?• _______, ________• _______, ________• : ≠ or < or >The TI-84 Z-Test output gives us a p-value of ________.Therefore we { reject or fail to reject } H0.Draw a bell-curve diagram of the story:






Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, s Known)

Draw a bell curve diagram for this problem.

Could you calculate the z test statistic and find the critical z value if the problem asked for those, too?

Excel’s Z.TEST(data array, μ, σ) function

• Give it the cells (“range”) of the sample data values.• And the μ mean from the null hypothesis.• And the σ population standard deviation.• It gives you back the one-tailed p-value.

• So it calculates a z = test statistic based on the sample.

• And then it calculates the area in the tail beyond that z

• Either left tail beyond negative z• Or right tail beyond positive z, it knows which one.

Excel’s Z.TEST(data array, μ, σ) function

• Give it the cells (“range”) of the sample data values.• And the μ mean from the null hypothesis.• And the σ population standard deviation.• It gives you back the one-tailed p-value.• You compare that p-value to the α level of

significance that was pre-selected in the design.• If p-value < α, then reject H0.• If p-value > α, then fail to reject H0.

Excel’s Z.TEST(data array, μ, σ) if you have a Two-Tailed Situation

• Recall the two-tailed situation: Ha has a ≠.• So the rejection regions are two tails of area α/2.• Excel Help says to do this to get the p-value:

2 * MIN(Z.TEST(data , μ, σ), 1 – Z.TEST(data , μ, σ))(whereas we tell TI-84 it’s two-tailed and TI-84 takes care of doubling it transparently, thank you.)

• And then decide as usual:• If p-value < α, then reject H0.• If p-value > α, then fail to reject H0.

Section 10.2

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