Download Lab 8 for Math 17: Hypothesis Testing II 1 Hypothesis Testing and

Lab 8 for Math 17: Hypothesis Testing II
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1.1
Hypothesis Testing and Confidence Intervals in Rcmdr
Proportions - One or Two
Rcmdr can do these if your data is entered in a special format - zeros and ones for responses. Since
that is generally not how this data is summarized for you, we will continue doing these by hand.
A bit more detail is in the help guide online or you can ask me if you really want to try it using
Rcmdr.
1.2
Means - One Mean
To perform a one-sample t-test or get a CI for µ, under the Statistics menu, select Means and then
select Single- sample t-test. In the window that opens, select the variable of interest. Then, you
need to set several options: the value of µ0 in the null hypothesis, whether your test is 2-sided
(first option) or one-sided to a certain direction (second and third options), and what confidence
level you want (for CIs). When you have selected all options and a variable, click Ok. Note that
there are defaults for the procedure, namely it will test H0 : µ = 0 vs. HA : µ 6= 0 and use 95%
confidence for CIs.
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Water Temperatures
Water from a power plant used for cooling is discharged into a river. Previous research has found
that the ecosystem will not be negatively affected if the mean temperature of the discharged water
is at most 150 degrees F. Regulators want to verify that the plant is following the 150 degree
guideline so they obtain the temperatures for a random sample of 50 discharged water samples.
The data was collected in waterpowertemp.dat. You may use R/Rcmdr for any/all parts below.
Perform a preliminary analysis of the water temperatures. What do you see?
What hypotheses should you test to help the regulators? Be sure to define your parameter.
What conditions do you need to check? Do they check out?
Report the test statistic and p-value for your test.
What conclusion do you reach?
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Interpret the test statistic after verifying its computation by hand.
Provide a numerical value for the standard error of the sample mean. Interpret this value.
If you had decided to see if the average water temperature was less than 150 degrees F (as an
alternative hypothesis), how could you know very quickly that you would have a large p-value?
(General) Consider the following: A hypothesis test results in a p-value of .0348. At α = .05, the
null is rejected. At α = .04, the null is rejected. At α = .035, the null is rejected. At α = .0348, the
null is rejected. At any α < .0348, the null is not rejected. Therefore, the p-value is the smallest
largest significance level at which the null hypothesis is rejected (choose one).
Explain why it was more appropriate to use a hypothesis test than a confidence interval to address
the issue the regulators were concerned about.
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Ear Infections
(Data from Uhari 1998, discussed in Utts/Heckard) Preschool children participated in a study to
test the effectiveness of the sweetener Xylitol in preventing ear infections. 165 children received
placebo daily doses and 68 of them got an ear infection during the study (three month duration).
Another 159 children received Xylitol daily doses and 46 got an ear infection during the study. The
study was a randomized experiment.
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a. Obtain a 90 percent confidence interval for the difference between proportions of ear infections
between placebo and Xylitol groups. Be sure to check assumptions.
b. Interpret your interval.
c. Interpret the confidence level.
d. If someone told you that one of the assumptions for this CI was that the populations were
normally distributed, how would you explain that assumption does not make sense?
e. Would you be able to conclude that there is no difference between the placebo and Xylitol groups
based on the CI? What significance level is associated with that decision? Would you be able to
conclude that Xylitol is more effective than placebo at reducing the risk of ear infection? What
significance level is associated with that decision? Explain.
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f. Now, perform a hypothesis test to test whether Xylitol is more effective than placebo at reducing
the risk of ear infection. Choose a significance level that makes the test equivalent to your confidence interval as determined in e. Verify that you reach the same conclusion with your hypothesis
test as with the CI.
g. What is the distribution of the test statistic assuming the null hypothesis is true? (I.E. What
distribution did you use to find the p-value?)
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Correcting a Hypothesis Test
You stumble upon a crumbled piece of paper that appears to have been dropped by another intro
stats student. The paper has a partial homework problem on it, but the context is lost as it is just
their scratchwork. Based on the scratchwork below, can you help them correct any issues before
they make their final writeup? List your corrections on another sheet.
1-sample t-test for population mean
1. Hypotheses: H0 : µ = 3200 vs. HA : x̄ < 3200
2. Conditions: RS is stated. Need to check ten successes and ten failures. ???
(previous line has been crossed out and erased a few times)
3. Mechanics. x̄ = 3100 and s = 250 with an n = 100.
z=
x̄ − µ0
100
3100 − 3200
√ =
=
=4
250/10
25
s/ n
Use t distribution with 99 df for p-value. 99 not in table so round up to df=100.
p-value is greater than 1-.005, so greater than .995, which is huge.
4. Conclusion (Decision): (Do not reject null hypothesis). We do not have evidence to conclude
the mean crop output has decreased from last year’s average of 3200.
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