# Stat 200 spring 2016 quiz 3

The following information is for Questions 1 through 6.

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The recommended daily allowance (RDA) of cobalamine (Vitamin B12) for growing teens is 2.4 µg (micrograms).  My brother takes it upon himself to make sure that everyone gets the message.  It is generally believed that growing teens are getting less than the RDA of 2.4 µg of cobalamine daily.

It is an open secret that Au & Associates FauxPharmaceutical (FAP) peddles dietary supplements around the globe.  It is claimed by representatives of FAP that by taking their vitamin supplement extracted from Atlantic scrod, brand-named as Pillule-au-Sashimi, teens will have the RDA of cobalamine, namely that µ = 2.4.  Médecins Sans Frontières (MSF) has a strong contingent of volunteers in the People’s Republic of Banana.  It is in this remote republic MSF is going to take on FAP for good.

MSF volunteers managed to collect with a 24-hour period blood samples of 10 randomly selected teens in the People’s Republic of Banana for cobalamine.  The amounts of cobalamine (in µg) determined in these 10 randomly selected teens are given as follow:

1.85    2.35    1.87    1.90    1.37    2.35    2.55    2.28    1.95    2.49

Based on their global experience, MSF assumes that the population standard deviation of cobalamine in teens to be σ = 0.56 µg.

Now, you are asked to weigh in on the dispute between Médecins Sans Frontières (MSF) and Au & Associates FauxPharamaceutical (FAP).

1.  (3 points)  Given the above information, what kind of hypothesis test will you conduct?  z-test, t-test, or, χ2-test?   Please explain.

2.  (3 points)

(a) What will be the null hypothesis?  H0 :

(b) What will be the alternative hypothesis?  H1 :

(c) What is the “tailedness” of the test (left-tailed, right-tailed, or two-tailed)?

3.  (10 points)

(a) What is the mean of the sample xbar?

(b) What is the standard deviation of the sample s1?

(c) What is the standard deviation of the population σ?

(d) What is the size of the sample n?

(e) Compute the z-score using xbar, µ, σ, and n.

4.  (5 points)  What is the corresponding p-value of the hypothesis test?

Use the Z table – Normal Distribution Calculator to calculate p. Attach a screen shot showing how you got your p value.

5.  (15 points)  The standard levels of confidence used in disputes are 95% (α = 5%) and 99% (α = 1%).

(a) Suppose a 95% confidence level is chosen. Will the null hypothesis be accepted or rejected?

(b) Suppose a 99% confidence level is chosen. Will the null hypothesis be accepted or rejected?

(c) Who would win the dispute based on a 95% confidence interval? (FAP or MSF)

6.  (10 points)  But, wait.  What if MSF actually does not know the population standard deviation? Then we should compute the t-statistic for our analysis.

(a) Compute the t-statistic using xbar, µ, s, and n.

(b) What is the number of degrees of freedom df?

(c) Use the T Distribution Calculator to determine the p-value in this case. Attach a screen shot to show how you got your answer.

(d) Will the null hypothesis be accepted or rejected if a confidence level of 99% is used?

The following information is for Question 7.

The tête-à-tête between MSF and FAP broke down, as anyone would have anticipated.  They are going to court.

To prepare for the upcoming court case, FAP managed to get hold of the same 10 teens randomly selected by MSF, and performed its own testing on them.  Their corresponding findings on the teens, in the same order as listed by MSF, are listed below.

2.45    2.85    2.87    2.32    1.98    2.51    1.75    1.98    2.03    2.89

It is understandable that FAP would have findings with a higher level of cobalamine in the teens.  But it is important to authenticate FAP’s findings.

It is imperative to make sure that FAP actually have the same group of teens tested by MSF.  His Honor, Judge Ig Sushi is presiding over this landmark case.  Unexpectedly, Professor Au is deemed to be able to act impartially, and appointed by His Honor to be the amicus curiæ in this pending court case.  (As a reward for his faithful service, Professor Au will be offered lifelong supply of bananas from the republic.)

After a few tasty bananas, Professor Au suggested to His Honor that the court should perform a statistical test to see if there is any difference in the means of the MSF group and the FAP group based on a court chosen significance level.  Furthermore, Professor Au informed His Honor that it would be unlikely that either MSF or FAP actually knew the population standard deviation at all.

7.  (16 points)  Perform the statistical test on the differences recommended by Professor Au.  Start by filling in the following table

 MSF 1.85 2.35 1.87 1.90 1.37 2.35 2.55 2.28 1.95 2.49 FAP 2.45 2.85 2.87 2.32 1.98 2.51 1.75 1.98 2.03 2.89 Difference

(a)    What is the mean of the differences dbar?

(b)   What is the standard deviations of the differences s?

(c)    What is the null hypothesis H0?

(d)   What is the alternative hypothesis H1 ?

(e)    Is this a 1tail or 2 tail test?

(f)    Find the t-statistic.

(g)   What is the number of degrees of freedom df?

(h)   Use the T Distribution Calculator to determine p-value in this case. Attach a screen shot showing how you got your answer.

(i)     Will the null hypothesis be accepted or rejected using a 95% level of confidence?

(j)     Who wins the court case? (FAP or MSF)

The following information is for Questions 8 through 10.

For the sake of argument, let us assume that His Honor, Judge Sushi intends to rule in favor of MSF.  However, he is rather apprehensive of errors in any statistical testing, especially with respect to Type I and Type II error.  Let us assume he picks a 95% significance level to be the benchmark of this court case.

8.  (3 points)

(a) What is a type I error in this case when H0: μ = 2.4 and H1: μ < 2.4 ?

(b) What is the probability of a type I error in this case?

9.  (10 points)  What if the RDA of cobalamine is actually not well established in the medical community?  In fact, it may span a range of values in µg, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6.  What is the corresponding Type II error and power for each of the given assumed RDA value?

Use the Z table – Normal Distribution (http://davidmlane.com/hyperstat/z_table.html) applet for the following problem.

Let the Mean = 2.4 and the SD = 0.1216. Find p for all values below 2.2. (Note: you should get the area/probability = 0.05 with the left side shade black. If you do not get this, do not go any farther in doing this problem!) The black area represents the region of p where we will reject the null hypothesis and instead accept the alternative hypothesis. The white area represents the region of p where we will accept the null hypothesis and reject the alternative hypothesis.

For our test, if the sample mean xbar is below 2.2, then we will reject H0. If the sample mean xbar is above 2.2, then we will accept H0.

A type I error is when we “reject the truth” of H0, i.e. the sample mean xbar is below 2.2 (in the black area) by chance and μ is actually equal to 2.4.

A type II error is when we “accept the falsehood” of H0, i.e. the sample mean xbar is above 2.2 (in the white area) by chance and μ is actually less than 2.4.

(a)    Calculate z = (xbar – μ)/SD for the following table.

(b)   Calculate the corresponding β value using z-table normal distribution applet. Do this by changing the mean, keeping the SD constant, and finding the area above 2.2.

(c)    Calculate the power (1 – β).

(d)   What is the meaning of power?

 Alternative True Mean, μ z β 1 – β 2.0 2.1 2.2 2.3 2.4 2.5 2.6

10.  (5 points)  Plot the power curve vs the alternative true mean μ.  What do we observed in this plot?  Please explain how this plot helps us to understand our decision to either accept or reject the null hypothesis.

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