Question 1

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed, with mean and standard deviation ?= 2.4. You select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days, are as follows:
63 $\quad$69 $\quad$62 $\quad$66 Based on these data, what is the 99% confidence interval for ?, in days?

Accepted Answer

The sample mean is 65.00. For a 99% confidence interval with a sample size of 4, use the t-distribution with 3 degrees of freedom. The t-value is approximately 4.541. The margin of error is calculated as t-value * (standard deviation / sqrt(sample size)), which gives approximately 3.09.

Question 2

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed, with mean and standard deviation ?= 2.4. You select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days, are as follows:
63 $\quad$69 $\quad$62 $\quad$66 If you wanted the margin of error for the 99% confidence interval to be ±1 inch, which size of simple random sample should you select?

Accepted Answer

To achieve a margin of error (ME) of ±1 for a 99% confidence interval, we use the formula for the margin of error: $ME = z \times \frac{\sigma}{\sqrt{n}}$, where $z$ is the z-score corresponding to the confidence level (for 99%, $z \approx 2.576$), $\sigma$ is the standard deviation (2.4 in this case), and $n$ is the sample size. Rearranging the formula to solve for $n$, we get $n = \left(\frac{z \times \sigma}{ME}\right)^2$. Plugging in the values, $n = \left(\frac{2.576 \times 2.4}{1}\right)^2 \approx 38.06$, which rounds up to 39 to ensure the margin of error is not exceeded.

Question 3

Your advisor has asked you to reduce the margin of error on your confidence interval from ±5 mg/l to ±1 mg/l. The population standard deviation is known to be 8 mg/. What should you do?

Accepted Answer

The answer of Your advisor has asked you to reduce...

Question 4

The level of nitrogen oxides (NO_x) in the exhaust of cars of a particular model varies Normally, with standard deviation σ= 0.05 gram per mile (g/mi). Government regulations call for NO_x emissions no higher than 0.3 g/mi. In testing H₀: μ= 0.3 versus H_a: μ< 0.3 using significance level 0.05, we find that the power of this test against an alternative of μ= 0.28 based on a random sample of size n = 40 is 0.808. What impact would using a larger sample size (n = 50) have?

Accepted Answer

Increasing the sample size generally increases the power of a hypothesis test, as it provides more information and reduces the effect of random variation. Therefore, with a larger sample size of 50, the power of the test against an alternative of μ= 0.28 would likely increase. The probability of a Type I error (rejecting the null hypothesis when it is actually true) should remain the same as the significance level is fixed at 0.05.

Question 5

People tend to give more socially acceptable answers when asked a sensitive question in person rather than anonymously via computer. When computing a 95% confidence interval for a population parameter, what does the margin of error cover?

Accepted Answer

The margin of error in a confidence interval only covers the statistical variation due to sampling error and does not cover the bias due to non-random sampling or the bias due to respondents giving socially desirable answers. Therefore, option D is the correct choice.

Question 6

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value of the parameter of interest. What is the probability that the researcher will commit a Type I error?

Accepted Answer

The probability of committing a Type I error is equal to the level of significance, which is α=0.05. This value is given in the question and is not affected by the power of the test or the alternative value of the parameter of interest.

Question 7

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value for the parameter of interest. What is the probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power?

Accepted Answer

The probability of committing a Type II error is equal to the power of the test. Since the power is given as 0.80 in the problem statement, the probability of committing a Type II error is 0.20 (1 - power). Therefore, the correct answer is B.

Question 8

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

Accepted Answer

Rejecting the null hypothesis when it is actually true is known as committing a Type I error. This means that we have incorrectly concluded that there is a significant effect or difference when there is not, and we have made a false positive claim. The level of significance (alpha) that we choose for our hypothesis test determines the probability of making a Type I error.

Question 9

When performing a test of hypotheses, what is a Type II error?

Accepted Answer

A Type II error occurs when we fail to reject the null hypothesis when it is actually false. This means we are accepting a false premise and may miss identifying a relationship or effect that actually exists.

Question 10

Which of the following will increase the value of the power in a statistical test of hypotheses?

Accepted Answer

Increasing the sample size will increase the value of power in a statistical test of hypotheses by decreasing the standard error and increasing the chance of detecting a real difference between two groups. Option A would decrease the power, and option C is incorrect because rejecting the null hypothesis at a lower level of significance would actually decrease the power. Therefore, the best choice is B.

Question 11

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

Accepted Answer

The assumption that a population mean is known is not included in the simple conditions for inference about a mean. The simple conditions assume simple random sampling with no nonresponse or other practical difficulty, a response variable with a Normal distribution in the population, and a population standard deviation that is known or can be estimated from the sample.

Question 12

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?

Accepted Answer

Using a larger confidence level increases the width of the confidence interval because it requires capturing more of the population mean with greater certainty, thus expanding the margin of error.

Question 13

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?

Accepted Answer

The quote refers to the possibility that the statistically significant results are a Type I error, meaning that the observed effect (tumor rates) is due to chance rather than a true effect of the treatment. This is because the rats used in the study are known to be prone to tumors, so the high tumor rates in some groups could be purely coincidental.

Question 14

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?

Accepted Answer

The quote suggests that the failure to observe significant tumor clusters in the control groups might be due to the inability to detect an effect (e.g., tumor development) that is actually present, which is the definition of a Type II error.

Question 15

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.

Accepted Answer

The answer of A study examined the link between sodium...

Question 16

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?

Accepted Answer

With a larger sample size, the margin of error decreases. Therefore, if there were more than 636 participants, the resulting margin of error for a 95% confidence interval would have been smaller than the original margin of error.

Question 17

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?

Accepted Answer

A 99% confidence interval would have a larger margin of error than a 95% confidence interval, as the higher confidence level requires a wider interval to be constructed.

Question 18

The makers of the pomegranate juice POM Wonderful claimed in advertisements that the juice helps with erectile dysfunction; this claim was based on a study showing a significant improvement after drinking POM Wonderful. However, the Federal Trade Commission (FTC) filed a formal complaint "alleging that the erectile dysfunction claims are false and unsubstantiated because the study on which the company relied did not show that POM Juice was any more effective than a placebo." What can we say about the FTC complaint?

Accepted Answer

The FTC complaint reflects the fact that statistical significance (which the study did show) is not the same as practical significance (which the study did not show). The claim made by POM Wonderful was based solely on statistically significant results from a single study, rather than on any practical benefits demonstrated by the juice.

Question 19

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. Based on these data, what is the value for the margin of error for a 95% confidence interval for μ?

Accepted Answer

The margin of error for a 95% confidence interval can be calculated using the formula: 
Margin of Error = 1.96 * (σ/√n), where σ is the standard deviation and n is the sample size. 
In this case, σ = 0.3 and n = 4. 
Plugging these values into the formula, we get: 
Margin of Error = 1.96 * (0.3/√4) = 0.294 
Therefore, the margin of error is 0.294 kg, which corresponds to choice B.

Question 20

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?

Accepted Answer

The margin of error decreases with the square root of the sample size. Increasing the sample size from 4 to 16 quadruples the sample size, so the margin of error would be halved.

Quiz 14: Exercises