Exam 14: Exercises

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams, mg) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is less than advertised. To explore this question, you test the following hypotheses: H₀: μ= 10, H_a: μ< 10 You carry out the test at a significance level of 0.05 by randomly selecting 100 people with cholesterol levels more than 200 mg/dl and, after they have eaten this brand of oatmeal for one month, computing the mean decrease x̄ in the subjects' cholesterol levels. What is the probability of a Type I error for your test at μ= 8.5?

A researcher proposes a study to investigate a quantitative variable for which the population standard deviation is known to be 4. She wants a margin of error of 2.5 with 95% confidence. Which sample size should she use?

Because small random samples have a lot of chance variation, even large deviations from the null hypothesis can fail to be statistically significant if the sample is small.

What does statistical significance depend on?

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. If there had been more than 636 participants in this group, how would the resulting margin of error for a 95% confidence interval compare?

Does 30 minutes of meditation every day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who meditated for 30 minutes each day for a period of 6 months. At the end of the study, 300 variables related to the mental performance of the subjects were measured on each subject, and the means compared to known means for these variables in the population of all adults. Sixteen of these variables were significantly better (in the sense of statistical significance) at the α= 0.05 level for the group who performed 30 minutes of meditation each day as compared to the population as a whole, and 3 variables were significantly better at the α= 0.01 level for the group who performed 30 minutes of meditation each day as compared to the population as a whole. What would be the correct conclusion?

The P-value gives the probability that the data are the result of chance variations alone.

The z procedures and many other procedures designed for Normal distributions rely Normality of individual observations in the population.

Which of the following is NOT an assumption included in the simple conditions for inference about a mean?

The pH measurements of water specimens from various locations along a given river basin are Normally distributed, with standard deviation σ= 0.3. The average pH of water specimens from 4 randomly selected locations on this river basin is x̄= 7.8. A 95% confidence interval for μ was computed for these data. If, instead of 4 water specimens, we had measured the pH of 16 water specimens, how would the margin of error of a 95% confidence interval for μ compare to the original 95% confidence interval based on a sample of 4 specimens?

A study randomly assigned 200 lab rats to 20 different groups: 2 control groups and 18 treatment groups in which the rats were fed various amounts of genetically modified corn exposed to the pesticide Roundup. The researchers ran a separate hypothesis test for each group and found a significant (P-value < 0.05) number of tumors for 2 treatment groups. The published findings received some criticism. Here is a quote about this study: "[T]he fact that such clusters of tumors didn't show up in the two small control groups could easily be due to random chance." What does this quote refer to?

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. Which sample size, n, would be required to obtain a margin of error of 0.01 kg for a 95% confidence interval for the mean total body protein of adult men with liver cirrhosis?

A radio show runs a phone-in survey each morning. One morning the show host asked listeners whether the city should change regulations for new structures at local beachfront property to save certain sea grass populations. Most of those phoning in their responses answered, "No, it is just grass-there is no reason to worry about it," and the station reported the results as statistically significant. What may we safely conclude?

What is the consequence of rejecting the null hypothesis when, in fact, it is true?

You collect a random sample of size n from a population; from the data collected, you compute a 95% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data?

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. The numerical value of the margin of error for this confidence interval is __________ mm Hg.

A researcher wishes to determine if aerobic exercise improves mental performance immediately following the exercise. He plans to have high school students participate in 30 minutes of aerobic exercise and then take a standard test of their reasoning skills. Suppose the scores of high school students on this test of reasoning skills immediately after 30 minutes of aerobic exercise follow a Normal distribution, with mean μ and standard deviation σ= 4. Suppose also that, in the general population of all high school students, scores on the test of reasoning skills follow a Normal distribution, with mean 25 and standard deviation σ= 4. The researcher, therefore, decides to test the following hypotheses: H₀: μ= 25, H_a: μ> 25 To do so, the researcher has 10,000 high school students perform 30 minutes of aerobic exercise and then, immediately following the exercise, take the test. The mean score for these students is x̄= 25.2 and the P-value is less than 0.0001. It is appropriate to conclude which of the following?

A study randomly assigned 200 lab rats to 20 different groups: 2 control groups and 18 treatment groups in which the rats were fed various amounts of genetically modified corn exposed to the pesticide Roundup. The researchers ran a separate hypothesis test for each group and found a significant (P-value < 0.05) number of tumors for 2 treatment groups. The published findings received some criticism. Here is a quote about this study: "Critics point out that the type of rats used in this study are particularly prone to tumors. So if you divide up 200 of them into 20 groups, you are likely to get very high (statistically significant) tumor rates in some of the groups." What does this quote refer to?

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. How would the margin of error for a 99% confidence interval based on the same 636 study participants compare?

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 67 randomly selected adult males with liver cirrhosis, has a margin of error of 0.024 kg. If the study had enrolled more subjects, how would the resulting margin of error for a 95% confidence interval compare?