Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.
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Transcript of Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.
![Page 1: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/1.jpg)
Ch. 10 examplesPart 1 – z and t
Ma260notes_Sull_ch10_HypTestEx.pptx
![Page 2: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/2.jpg)
Test on a single population mean• Intro: So far we’ve studied estimation, or confidence
intervals, focusing on the middle 90% or 95% or 99%• Now the focus is on Hypothesis Testing (using the 1%
or 5% or 10% in the tails instead)
• As in Ch. 8 (CLT) and 9 (CI s), we’ll study both:– Means and – proportions
![Page 3: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/3.jpg)
Courtroom Analogy-Two Hypotheses: Guilty or Not Guilty
Our conclusion (the sentence)
Verdict is Not guilty Verdict is Guilty
Reality Defendant is innocent
Acquittal- Correct conclusion
Error
Defendant is guilty
Error Conviction- Correct conclusion
![Page 4: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/4.jpg)
Hypothesis Testing in Statistics—Note the chance of two different errors (Type I, II)
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Types of Hypotheses in Statistical Testing• The null hypothesis H0 is one which expresses the
current state of nature of belief about a population– Note: The hypothesis with the equal sign is the null
• The alternative (or research) hypothesis (H 1 or H
a ) is one which reflects the researcher’s belief. (It will always disagree with the null hypothesis).– Note: the alternative hypothesis can be one or two tailed.– In one tailed tests, the alternative is usually the claim
![Page 6: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/6.jpg)
Summary of Hypothesis test steps1. Null hypothesis H0, alternative hypothesis H1, and preset α
2. Test statistic and sampling distribution
3. P-value and/or critical value4. Test conclusion
5. Interpretation of test results
2-tailed exH0: µ= 100H1: µ ≠ 100α = 0.05
Left tail exH0: µ = 200H1: µ < 200α = 0.05
Right tail exH0: µ = 50H1: µ > 50α = 0.05
CV approach P value approach decisionIf test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 7: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/7.jpg)
Should you use a 2 tail, or a right, or left tail test?
• Test whether the average in the bag of numbers is or isn’t 100.
• Test if a drug had any effect on heart rate.• Test if a tutor helped the class do better on the
next test.• Test if a drug improved elevated cholesterol.
2-tailed exH0: µ= __H1: µ ≠ __
Left tail exH0: µ = ___H1: µ < ___
Right tail exH0: µ = __H1: µ > __
![Page 8: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/8.jpg)
Example #1- numbers in a bagRecall the experiments with the bags of
numbers.• I claim that µ = 100. We’ll assume =21.9. • Test this hypothesis (that µ = 100 )if your
sample size n= 20 and your sample mean was 90.
![Page 9: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/9.jpg)
Ex #1- Hyp Test- numbers in a bag1. H0: µ = 100
H1: µ ≠ 100
α = 0.05
2. Z = =
3. CV
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decision
If test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 10: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/10.jpg)
Ex #2– new sample mean
• If the sample mean is 95, redo the test:1. H0: µ = 100
H1: µ ≠ 100
α = 0.05
2. Z = =
3. CV
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decisionIf test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 11: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/11.jpg)
Ex #3: Left tail test- cholesterol
• A group has a mean cholesterol of 220. The data is normally distributed with σ= 15
• After a new drug is used, test the claim that it lowers cholesterol.
• Data: n=30, sample mean= = 214.
![Page 12: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/12.jpg)
Ex #3- cholesterol – left test1. H0: µ 220 (fill in the correct hypotheses here)
H1: µ 220
α = 0.05
2. Z = =
3. P-value and/or critical value
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decision
If test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 13: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/13.jpg)
Ex #4- right tail- tutor
Scores in a MATH117 class have been normally distributed, with a mean of 60 all semester. The teacher believes that a tutor would help. After a few weeks with the tutor, a sample of 35 students’ scores is taken. The sample mean is now 62. Assume a population standard deviation of 5. Has the tutor had a positive effect?
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Ex #4: tutor1. H0: µ 60
H1: µ 60
α = 0.05
2. Z = =
3. P-value and/or critical value
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decision
If test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 15: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/15.jpg)
When is unknown…use t tests
Just like with confidence intervals, if we do not know the population standard deviation, we– substitute it with s (the sample standard
deviation) and– Run a t test instead of a z test
First, we’ll review using the t table
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Critical values for t• Find the CV for RIGHT tail examples when:
– n=10, = .05– n=15, = .10
• Find the CV for LEFT tail examples when:– n=10, = .05– n=15, = .10
• Find the CV for TWO tail examples when:– n=10, = .05– n=15, = .10– n=20, = .01
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Ex #5– t test – placement scores
The placement director states that the average placement score is 75. Based on the following data, test this claim.
• Data: 42 88 99 51 57 78 92 46 57
![Page 18: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/18.jpg)
Ex #5 t test – placement scores
1. H0: µ 75
H1: µ 75α = 0.05
2. t = =
3. P-value and/or critical value4. Test conclusion
5. Interpretation of test results
6. Confidence Interval
CV approach P value approach decision
If test stat is in RR If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 19: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/19.jpg)
Ex #6- placement scores
The head of the tutoring department claims that the average placement score is below 80. Based on the following data, test this claim.
• Data: 42 88 99 51 57 78 92 46 57
![Page 20: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/20.jpg)
Note: P values for t-tests
• We can use the t-table to approximate these values
• Use “Stat Crunch” on the online-hw to get more exact answers. – See the rectangle to the right of the data
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Ex #6– t example
1. H0: µ 80 (fill in the correct hypotheses here)
H1: µ 80α = 0.05
2. t = =
3. P-value and/or critical value
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decision
If test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR
p-value > α do not reject H0
![Page 22: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/22.jpg)
Ex #7- salaries– t
A national study shows that nurses earn $40,000. A career director claims that salaries in her town are higher than the national average. A sample provides the following data:
41,000 42,500 39,000 39,99943,000 43,550 44,200
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Ex #7- salaries
1. H0: µ 40000 (fill in the correct hypotheses here)
H1: µ 40000α = 0.05
2. t = =
3. P-value and/or critical value
4. Test conclusion
5. Interpretation of test results
CV approach P value approach decision
If test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR
p-value > α do not reject H0
![Page 24: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/24.jpg)
The Hypothesis Testing templateAlthough hw may be worded differently, questions on the test will
require you to do a 5 or 6 step answer (using either p value or CV)
1. Null and alternative hypotheses, H0 and H1, and preset α
2. Test statistic z = or t =3. P-value and/or critical value
4. Test conclusion– Reject H0 or notIf test value is in RR (p-value ≤ α), we reject H0.
If test value is not in RR (p-value > α), we do not reject H0
5. Interpretation of test results6. Confidence Interval (for 2 tailed problems)
![Page 25: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/25.jpg)
Ch.10 – part twoTesting Proportion p
• Recall confidence intervals for p:
• ± z
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Hypothesis tests for proportions1. Null hypothesis H0, alternative hypothesis H1, and preset α
2. Sampling distribution Test statistic z =
3. P-value and/or critical value
4. Test conclusion
5. Interpretation of test results
2-tailed exH0: p= .5H1: p ≠ .5α = 0.05
Left tail exH0: p = .7H1: p < .7α = 0.05
Right tail exH0: p = .2H1: p > .2α = 0.05
CV approach P value approach decisionIf test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 27: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/27.jpg)
Ex #8- proportion who like job
• The HR director at a large corporation estimates that 75% of employees enjoy their jobs. From a sample of 200 people, 142 answer that they do. Test the HR director’s claim.
![Page 28: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/28.jpg)
Ex #81. Null hypothesis H0, alternative hypothesis H1, and preset α
H0: p=.75
H1: p
α = 2. Test statistic and sampling distribution
Z = =
3. P-value and/or critical value4. Test conclusion
5. Interpretation of test results6. Confidence interval
CV approach P value approach decisionIf test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0
![Page 29: Ch. 10 examples Part 1 – z and t Ma260notes_Sull_ch10_HypTestEx.pptx.](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e0d5503460f94af6ff5/html5/thumbnails/29.jpg)
Ex #9
Previous studies show that 29% of eligible voters vote in the mid-terms. News pundits estimate that turnout will be lower than usual. A random sample of 800 adults reveals that 200 planned to vote in the mid-term elections. At the 1% level, test the news pundits’ predictions.
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Ex #91. Null hypothesis H0, alternative hypothesis H1, and preset α
H0: p (fill in hypothesis)
H1: p
α = 2. Test statistic and sampling distribution
Z = =
3. P-value and/or critical value4. Test conclusion
5. Interpretation of test results6. Confidence Interval
CV approach P value approach decisionIf test stat is in RR (Rejection Region)
If p-value ≤ α reject H0
If test stat is not in RR p-value > α do not reject H0