Biometry 109
Midterm II
Name:____________
3/29/06
(a.) Show all work to receive full credit.
(b.) Circle your final answer.
(c.) You may use Minitab (not email) and calculators. Explicitly describe any
commands used. No notes or internet help outside of Minitab.
(1.) Suppose men heights are normally distributed with a mean of 70 inches and a
standard deviation of 3 inches. 30 men are to be randomly sampled and their average
height calculated. Denote this average height as X . X is a random variable with a
certain mean and standard deviation.
(1a. 2pts) What is the mean of X ?
(1b. 4pts) What is the standard deviation of X ?
(1c. 6pts) Calculate P(69  X  71) . Show your work.
(2. 4pts) Suppose we have a random variable Y that has a slightly skewed distribution. A
large number, say 100, values of Y are going to be randomly sampled and the average of
those 100 Y values calculated producing Y . Keeping in mind the Central Limit
Theorem, circle which one of the following statements is true. That is, two of the three
statements are false and one is true, circle the true statement.
(i) The distribution of the Y random variable will become approximately normal
distributed after the sampling of those 100 values.
(ii) The distribution of the 100 sampled Y values will be approximately normally
distributed.
(iii) Y can be considered an approximately normally distributed random variable.
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(3.) A biologist planted 400 seeds of a new type of corn. 320 of the seeds germinated.
Assume independence between seeds and that each has the same probability of
germinating.
(3a. 6pts) Calculate a 95% confidence interval for the probability of a seed germinating
(population proportion). Show the formula or commands you used.
(3b. 4pts) Suppose the biologist would like to have the 95% confidence interval for the
probability of germination be no wider than 0.04 . What is the minimum sample size
needed? Show how you got this answer.
(4.) A January 2006 survey found that 57% of the American public believes that the
nation is "off on the wrong track." The margin of error was  3% . This is the equivalent
to a 95% confidence interval of 54-60%.
(4a. 4pts) What is implied to be within that interval of 54-60%?
(4b. 4pts) Explain what is meant by "95% confidence" with regards to a confidence
interval.
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(5.) The birth weights of 4 male infants and 5 female infants were recorded. Assume the
weights are independent of one another and from normal distributions. The weights are:
Male (grams)
2786
2317
2872
2715
Female (grams)
2866
2217
2293
2519
2671
(5a. 2pts) Use Minitab to calculate the mean and standard deviation of the male weights.
Show your commands.
(5b. 2pts) Calculate the mean and standard deviation of the female weights.
(5c. 4pts) Calculate the standard error for ymale  y female  . Show your work.
(5d. 6pts) Using 6 degrees of freedom, calculate the 95% confidence interval for
 male   female. Show your work, or describe the Minitab commands.
(5e. 4pts) Suppose we are interested in whether or not the mean birth weight of males is
greater than that of females. State the null and alternative hypotheses using the notation
 male   female.
(5f. 4pts) By hand, calculate the t-statistic for the hypothesis test. Show your work.
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(5g. 4pts) Suppose the p-value was 0.19. Using a level of significance of   0.05 , should
you keep or reject the null hypothesis?
(6. 6pts) A 2-sample t-test was performed to test H 0 : 1   2  0 against H A : 1   2  0 .
The data resulted in a t-statistic of 1.07 with 40 degrees of freedom. Calculate the pvalue. Show how you got your answer. If using the table, you may need to bracket your
answer.
(7. 6pts) Define p-value.
(8. 4pts) Define power.
(9.) Suppose a study was performed to investigate the difference between fertilizer X and
fertilizer Y on plant growth. The researchers hypothesized fertilizer Y would promote
more growth than fertilizer X. They fertilized 60 plants, 30 with Y and 30 with X. The
growth of plants with fertilizer Y was, on average, greater than those with fertilizer X, but
the p-value only came to 0.08, thus they concluded there was not statistically significant
evidence that fertilizer Y promoted a greater mean growth than fertilizer X.
(9a. 4pts) Suppose that, unknown to the biologists, fertilizer Y does promote more
growth. When planning the experiment, what is the primary thing the biologists could
have done to increase the power of the statistical test?
(9b. 4pts) Suppose that, unknown to the biologists, fertilizer & does promote more
growth. What type of statistical error was performed by the biologists? Type 1 or Type
2?
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(10. 4pts) Suppose a statistical test is performed and, unknown to the researchers, the null
hypothesis is true. What is the probability of rejecting the null hypothesis?
(11. 4pts) One of the following four statements is FALSE. Circle the false statement.
(i) An example of biological/practical significance is when the difference between
the effectiveness of two drugs is great enough so that one drug is preferable over
the other in treating the disease.
(ii) Even if the difference between two population means is very tiny, a large
enough sample size can make it very likely that the difference will be statistically
significant.
(iii) Regardless of the sample size, if there is no difference between the means
there will always be a probability of  of rejecting the null hypothesis that the two
means are equal.
(iv) When the alternative hypothesis is rejected we say that we have "statistically
significant evidence."
(12. 4pts) True or False: If the assumptions for a parametric test are valid, the appropriate
parametric test, such as the 2-sample t-test, will have more power the appropriate
nonparametric test, such as the Mann-Whitney test.
(13. 4pts) True or False: The calculation of a p-value is performed under the assumption
that the null hypothesis is true.
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