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Math& 146 - Discussion Forum for Module 13 Hypothesis Tests
Paleo Diet - Test of 2 Means
For the Discussion forum Subject Line write:
“Paleo Diet test” and then # the number of the step below that you will be presenting.
| |
|This problem is from Open Intro Statistics 2nd ed. book: |
|• The Paleo diet allows only for foods that humans typically consumed over the last 2.5 million years, excluding those agriculture-type foods that arose |
|during the last 10,000 years or so. |
|• Some people believe that following the Paleo diet will lead to more weight loss as compared to people who eat the foods more typical of current times in |
|this country. A study was conducted to check this claim. |
|• Researchers randomly divided 500 volunteers into two equal-sized groups. One group spent 6 months on the Paleo diet. The other group received a pamphlet |
|about controlling portion sizes. Randomized treatment assignment was performed, and at the beginning of the study, the average difference in weights between |
|the two groups was about 0. |
|• After the study, the Paleo group had lost on average 7 pounds with a standard deviation of 20 pounds while the control group had lost on average 5 pounds |
|with a standard deviation of 12 pounds. |
Note: a Test of Hypothesis is not normally numbered this way! These numbers are simply for the Discussion Forum assignment.
Use the 0.05 significance level in this test.
1. Is this situation about two Independent Samples or about Matched Pairs (dependent samples)? Explain why you decided as you did.
This is Independent Samples. The two groups (the two samples of 250 each) were randomly assigned. There were different people in the two samples, and they were not “matched” in any way.
2. State the null and alternative hypotheses in words (as sentences). Label which is which.
Null Hypothesis: There is no difference in mean weight loss from following the Paleo diet or following the typical diet of current foods . In other words: The average weight loss from the two diets is the same.
Alternative Hypothesis: People following the Paleo diet have greater average weight than do people following the diet of typical current foods.
3. State the null and alternative hypotheses in symbolic form. Use correct notation. Label which is which.
H0: [pic]Paleo = [pic]typical-current OR [pic]Paleo – [pic]typical-current = 0
Ha: [pic]Paleo > [pic]typical-current OR [pic]Paleo – [pic]typical-current > 0
ANOTHER WAY to answer this is to define sample 1 and 2 and then….
1 = Paleo and 2 = typical current
H0: [pic]1 = [pic]2 OR [pic]1 – [pic]2 = 0
Ha: [pic]1 > [pic]2 OR [pic]1 – [pic]2 > 0
4. Do the data meet the conditions for using our method of testing the hypothesis?
Clearly show why or why not. And make a conclusion: are the conditions met?
• the two samples are independent (were randomly assigned, are different people)
• each sample is random, and the sample size is greater than 30 (it is 250)
5. For this test , is the test statistic a Z distribution or T distribution or something else?
Explain how you know.
Write the Formula for the test statistic (NOT with numbers, just the general formula).
We use the T distribution since that is appropriate for two independent sample tests.
T = [pic]
6. Find the value of the test statistic, as a decimal value rounded to 4 decimal places.
Explain how you found it (formula used – with what numbers? Or calculator command – what one exactly?)
Using the calculator’s 2-SampTTest command, t = 1.3558
7.. Do NOT find the p-value in this step, but prepare to find it by:
a) Determine if this is a left-tail, right-tail, or 2-tail test. Right-tail
b) Shade the portion of the curve that is relevant
to finding the p-value.
c) And label relevant number(s) on the horizontal axis.
Shade tail to the right of 1.3558
0 1.3558
8. Find the p-value
a) Write the calculator command you use. 2-SampTTest command
b) State the p-value, rounded to 4 decimal places. P-value = .087954 ( .0880
9. a) Are the results significant? No
b) HOW do you know? (why did you decide that?) The p-value is greater than the significance level
Since .0880 > .05
c) State any and all of these that are correct conclusions:
Do not reject H0, Data do not support Ha
Reject H0 Do not reject H0 Accept H0 Data do not support Ha Data support Ha
10. Write a sentence or two in clear English stating the conclusion in the context of the problem. So, include words from the statement of the situation. E.g. in this case: weight loss, Paleo diet, foods typical of current times
The sample data are not strong enough to support the claim that the Paleo diet leads to greater weight loss than when people eat foods typical of current days.
Math& 146 - Discussion Forum for Module 12 Hypothesis Tests
Exam Scores – Test of 2 Means
For the Discussion forum Subject Line write:
“Exam Scores” and then # the number of the step below that you will be presenting.
|This problem is from Open Intro Statistics 2nd ed. book: |
| |
|• The National Center of Education Statistics conducted a survey of high school seniors, collecting test data on reading, writing, and several other subjects.|
|Here we examine a simple random sample of 200 students from this survey, and their scores on the reading and writing tests. |
|• The observed differences in scores (for reading score minus writing score) for each student was analyzed. For this sample, the mean was –0.545 and the |
|standard deviation was 8.887 points. |
|• The question being considered is whether the average reading score and average writing score are basically the same or not. Do these data provide |
|convincing evidence of a difference between the average scores on the two exams? |
Note: a Test of Hypothesis is not normally numbered this way! These numbers are simply for the Discussion Forum assignment.
Use the 0.05 significance level in this test.
1. Is this situation about two Independent Samples or about Matched Pairs (dependent samples)? Explain why you decided as you did.
This is Matched Pairs. The sample of people for the reading scores is exactly the same people as the sample for the writing scores (so the two samples depend on each other). And the scores are matched so that one person’s scores for the two tests are subtracted.
2. State the null and alternative hypotheses in words (as sentences). Label which is which.
Null Hypothesis: The average reading and writing scores are the same. OR There is no difference between the average reading and writing scores.
Alternative Hypothesis: The average reading and writing scores are not the same. OR There is a difference between the average reading and writing scores.
3. State the null and alternative hypotheses in symbolic form. Use correct notation. Label which is which.
Notation: [pic]d is the average of the differences between reading and writing scores.
H0: [pic]d = 0 OR [pic]reading – [pic]writing = 0
Ha: [pic]d [pic] 0 OR [pic]reading – [pic]writing [pic] 0
4. Do the data meet the conditions for using our method of testing the hypothesis?
Clearly show why or why not. And make a conclusion: are the conditions met?
• the sample was random and the sample size is greater than 30 (it is 200)
5. For this test , is the test statistic a Z distribution or T distribution or something else?
Explain how you know. A T distribution is used for Matched Pairs (aka dependent samples) tests.
Write the Formula for the test statistic (NOT with numbers, just the general formula).
T = [pic]
6. Find the value of the test statistic, as a decimal value rounded to 4 decimal places.
Explain how you found it (formula used – with what numbers? Or calculator command – what one exactly?)
Calculate t = (–.545 – 0) / ( 8.887 / sqrt(200) ) = –.8673
OR use the calculator’s T-Test command
7.. Do NOT find the p-value in this step, but prepare to find it by:
a) Determine if this is a left-tail, right-tail, or 2-tail test.
It is 2-tail test (since Ha has “not equal” sign)
b) Shade the portion of the curve that is relevant
to finding the p-value.
c) And label relevant number(s) on the horizontal axis. -.8673 0 +.8673
The two tails left of–.8673 and to the right of +.8673 are shaded
8. Find the p-value
a) Write the calculator command you use.
b) State the p-value, rounded to 4 decimal places.
Use calculator’s “tcdf” with degrees of freedom = n-1 = 200-1 = 199
Area of tail to the left = tcdf ( –9999999, –.8673, 199) = .193411
Area of tail to the right is equal to that.
P-value = area of 2 tails = 2 ( .193411) = .386822 ( p-value = .3868
OR use the calcualtor’s T-Test to get the same p-value.
9. a) Are the results significant? No
b) HOW do you know? (why did you decide that?)
The p-value is large. In particular it is greater than the significance level since .3868 > .05
c) State any and all of these that are correct conclusions:
Do not reject H0, Data do not support Ha
Reject H0 Do not reject H0 Accept H0 Data do not support Ha Data support Ha
10. Write a sentence or two in clear English stating the conclusion in the context of the problem. So, include words from the statement of the situation. E.g. in this case: reading scores, writing scores
There is not sufficient sample evidence to say that the reading and writing scores are different on average. We do not reject that the reading and writing scores are basically the same.
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