Sampling Distribution and Confidence Interval Problems



Assignment #5 - STAT 601 More Statistical Inference for μ and p

(44 points)

1) Type I and II Errors and Drug Safety Testing

An interesting paper on the subject can be found here at the link below. The material at the end of the paper gets a bit theoretical but the first few sections are very good.



Over the years, the U.S. Food and Drug Administration (FDA) has worked very hard to avoid making Type I errors.

Ho: Drug in question is ineffective or unsafe/dangerous

HA: Drug in question is effective and safe.

A type I error occurs when the FDA approves a drug that is not both safe and effective. Despite the agency’s efforts, however, bad drugs do on occasion get through to the public. For example, Omniflox, and antibiotic, had to be recalled less than six months after its approval due to reports of severe adverse reactions, which included a number of deaths. Similarly, Fenoterol, an inhaled drug intended to relieve asthma attacks, was found to increase the risk of death rather than decrease it.

a) Is there any way for the FDA to completely eliminate the occurrence of type II errors?

Explain. (2 pts.)

b) Name two ways the probability of making a type II error can be decreased. (2 pts.)

2) Power and Sample Size

Data from the Framingham Study ()

allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is μ ’ 219 mg/dl and the σ ’ 41 mg/dl. Suppose you wish to conduct a study to see if men who go on to develop coronary heart disease have a greater mean serum cholesterol level, i.e. you have the following hypotheses in mind when conducting your study using the Framingham data:

Ho: μ < 219 mg/dl

HA: μ > 219 mg/dl

You wish to conduct your test using an α ’ .05 level of significance. To be clear, in this study men had their serum cholesterol levels at the start of the study and then they were followed prospectively. Some of these men went on to develop coronary heart disease and some did not. We are comparing the initial cholesterol levels of these two groups of men.

a) State in words what a type I error is in this particular situation. (1 pt.)

b) State in words what a type II error is in this particular situation. (1 pt.)

c) What is the probability of making a type I error? (1 pt.)

d) Use the JMP DOE > Sample Size and Power calculator to find the power and the probability of making a type II error when n = 25 and the population mean making the alternative true is 244 mg/dl. (2 pts.)

e) Construct and turn in a copy of a plot of the power vs. the sample size when the population mean which makes the alternative true is equal to 244 mg/dl. Use it to estimate the sample size required to achieve a power of 90%. (3 pts.)

3)  In an effort to detect hypertension in young children, blood-pressure measurements were taken on 30 children aged 5-6 years living in a specific community.  For these children the mean diastolic blood pressure was found to be 56.2 mm Hg with standard deviation 7.9 mm Hg.  From a nationwide study, we know that the mean diastolic blood pressure is 64.2 mm Hg for 5- to 6-year-old children. 

a) Is there evidence that the mean diastolic blood pressure for children in this community is different from the nationwide average of children of the same age group? Use a 95% confidence interval for the mean answer this question. (4 pts.)

4) In the 1980s it was generally believed that autism affected about 5% of the nation’s children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. A recent study examined 384 children and found that 46 of them showed signs of some form autism

a) Is this strong evidence that the level of autism has increased? Conduct an appropriate test. (5 pts.)

b) Give a 95% CI for the proportion of children in the population who show signs of some form of autism and interpret this interval. (4 pts.)

5) Suppose we wanted to estimate the proportion of children in U.S. who exhibit signs of autism using a 95% CI with a margin of error no larger than 3%, i.e. E = .03.

a) What sample size is required if researchers are willing to assume that true proportion is at most 8%? (3 pts.)

b) What sample size is required if researchers are not willing to assume anything about this proportion? (3 pts.)

6) Suppose we wanted to estimate the mean systolic blood pressure of individuals more than 30% above their ideal body weight. How many such individuals would we have to sample to estimate this mean with a 95% confidence interval and margin of error no larger than 5 mmHg? Use your own prior knowledge of the range of systolic blood pressures to find an estimate of [pic] and use that in determining the sample size. (4 pts.)

7) The Bayley Scales of Infant Development yield scores on two indices – the Psychomotor Development Index (PDI) and the Mental Development Index (MDI) – which can be used to assess a child’s level of function in each of these areas at approximately one year of age. Among healthy infants, both indices have a mean value of 100. As part of study assessing the development and neurologic status of children who have undergone reparative heart surgery during the first three months of life, the Bayley Scales were administered to a sample n = 144 of one-year-old infants born with congenital heart disease.

These data are contained in the data file PDI-MDI.JMP contains the variables:

• PDI = psychomotor development index

• MDI = mental development index

a) Is there evidence that the mean PDI score for children born with congenital heart disease who undergo reparative heart surgery during the first three months of life have a mean score less 100, which is the mean for healthy infants? Summarize your findings and check assumptions. (4 pts.)

b) Is there evidence that the mean MDI score for children born with congenital heart disease who undergo reparative heart surgery during the first three months of life have a mean score different from 100, which is the mean for healthy infants? Summarize your findings your findings and check assumptions. (4 pts.)

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