### Math 1530 Quiz Complete Solution

**Math 1530 Quiz Complete Solution**

Question 1 (2.85 points)

Eggs that are contaminated with salmonella can cause food poisoning among consumers. A large egg producer takes an SRS of 200 eggs from all the eggs shipped in one day. The laboratory reports that 11 of these eggs had salmonella contamination. Unknown to the producer, 0.2% (two-tenths of one percent) of all eggs shipped had salmonella. In this situation,

Question 1 options:

a) both 0.2% and 11 are statistics.

b) 11 is a parameter and 0.2% is a statistic.

c) 0.2% is a parameter and 11 is a statistic.

d) both 0.2 % and 11 are parameters.

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Question 2 (2.85 points)

Suppose you interview 10 randomly selected workers and ask how many miles they commute to work. You'll compute the sample mean commute distance. Now imagine repeating the survey many, many times, each time recording a different sample mean commute distance. In the long run, a histogram of these sample means represents

Question 2 options:

a) a simple random sample.

b) the true population average commute distance.

c) the sampling distribution of the sample mean.

d) the bias, if any, that is present in the sampling method.

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Question 3 (2.85 points)

The distribution of actual weights of 8 oz wedges of cheddar cheese produced at a dairy is Normal with mean 8.1 ounces and standard deviation 0.2 ounces.

Reference: Ref 11-5

The probability that the average amount charged over these 160 procedures is more than $1180 is

Question 3 options:

a) 0.448.

b) 0.552.

c) 0.046.

d) 0.954.

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Question 4 (2.85 points)

In a large population of college-educated adults, the mean IQ is 112 with standard deviation 25. Suppose 300 adults from this population are randomly selected for a market research campaign.

Reference: Ref 11-4

The probability that the sample mean IQ is greater than 115 is

Question 4 options:

a) 0.452.

b) 0.528.

c) 0.019.

d) 0.981.

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Question 5 (2.85 points)

In a large population of college-educated adults, the mean IQ is 112 with standard deviation 25. Suppose 300 adults from this population are randomly selected for a market research campaign.

Reference: Ref 11-4

The distribution of the sample mean IQ is

Question 5 options:

a) approximately Normal, mean 112, standard deviation 1.443.

b) approximately Normal, mean 112, standard deviation 0.083.

c) approximately Normal, mean equal to the observed value of the sample mean, standard deviation 25.

d) approximately Normal, mean 112, standard deviation 25.

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Question 6 (2.85 points)

The central limit theorem says that when a simple sample of size n is drawn from any population with mean μ and standard deviation σ, then when n is sufficiently large

Question 6 options:

a) the distribution of the sample mean is approximately Normal.

b) the standard deviation of the sample mean is σ2/n.

c) the distribution of the population is approximately Normal.

d) the distribution of the sample mean is exactly Normal.

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Question 7 (2.85 points)

A survey of college students finds that 20% like country music, 15% like gospel music, and 10% like both country music and gospel music.

Reference: Ref 12-2

The proportion of students that like gospel music but not country music is

Question 7 options:

a) 75%.

b) 25%.

c) 20%.

d) 5%.

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Question 8 (2.85 points)

A survey of college students finds that 20% like country music, 15% like gospel music, and 10% like both country music and gospel music.

Reference: Ref 12-2

The proportion of students that like neither country music nor gospel music is

Question 8 options:

a) 25%.

b) 90%.

c) 40%.

d) 75%.

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Question 9 (2.85 points)

A survey of college students finds that 20% like country music, 15% like gospel music, and 10% like both country music and gospel music.

Reference: Ref 12-2

The proportion of students that like either country music or gospel music is

Question 9 options:

a) 25%.

b) 55%.

c) 75%.

d) 45%.

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Question 10 (2.85 points)

A survey of college students finds that 20% like country music, 15% like gospel music, and 10% like both country music and gospel music.

Reference: Ref 12-2

The conditional probability that a student likes country music given that he or she likes gospel music is

Question 10 options:

a) 2/3 (approximately 66.7%).

b) 50%.

c) 90%.

d) 15%.

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Question 11 (2.85 points)

Suppose we toss a coin and roll a die. Let A be the event that the number of spots showing on the die is three or less and B be the event that the coin comes up heads. The events A and B are

Question 11 options:

a) independent.

b) conditional.

c) reciprocals.

d) disjoint.

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Question 12 (2.85 points)

Spelling mistakes in a text are either nonword errors or word errors. A nonword error produces a string of letters that is not a word, such as “the” typed as “teh.” Word errors produce the wrong word, such as “loose” typed as “lose.” Nonword errors make up 25% of all errors. A human proofreader will catch 80% of nonword errors and 50% of word errors.

Reference: Ref 12-4

What percent of errors will the proofreader catch?

Question 12 options:

a) 37.5%

b) 80%

c) 20%

d) 57.5%

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Question 13 (2.85 points)

A level 0.95 confidence interval is

Question 13 options:

a) any interval with margin of error ± 0.95.

b) an interval with margin of error ± 0.95, which is also correct 95% of the time.

c) an interval computed from sample data by a method that has probability 0.95 of producing an interval containing the true value of the parameter of interest.

d) an interval computed from sample data by a method guaranteeing that the probability the interval computed contains the parameter of interest is 0.95.

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Question 14 (2.85 points)

A medical researcher treats 400 subjects with high cholesterol with a new drug. The average decrease in cholesterol level is = 90 after two months of taking the drug. Assume that the decrease in cholesterol after two months of taking the drug follows a Normal distribution, with unknown mean μ and standard deviation σ = 30.

Reference: Ref 14-1

Which of the following would produce a confidence interval with a smaller margin of error than the 95% confidence interval you computed above?

Question 14 options:

a) Give the drug to only 100 subjects rather than 400, since 100 people are easier to manage and control.

b) Give the drug to 500 subjects rather than 400.

c) Compute a 99% confidence interval rather than a 95% confidence interval. The increase in confidence indicates that we have a better interval.

d) None of the above

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Question 15 (2.85 points)

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed with mean μ and standard deviation σ = 2.4. I select a simple random sample of four plants of this variety and measure the time until maturity. The sample yields = 65.

A 95% confidence interval for μ, in days, is

Question 15 options:

a) 65.00 ± 3.95.

b) 65.00 ± 1.97.

c) 65.00 ± 4.70.

d) 65.00 ± 2.35.

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Question 16 (2.85 points)

The records of the 100 postal employees at a postal station in a large city showed that the average time these employees had worked for the postal service was = 9 years. Assume that we know that the standard deviation of the population of times U.S. Postal Service employees have spent with the postal service is approximately Normal, with standard deviation σ = 5 years. A 95% confidence interval for the mean time μ the population of U.S. Postal Service employees have spent with the postal service is

Question 16 options:

a) 9 ± 0.98.

b) not trustworthy.

c) 9 ± 9.80.

d) 9 ± 0.82.

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Question 17 (2.85 points)

The upper 0.05 critical value of the standard Normal distribution is

Question 17 options:

a) 1.960.

b) 2.576.

c) 1.645.

d) 2.326.

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Question 18 (2.85 points)

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

Question 18 options:

a) Use the same confidence level, but compute the interval n times. Approximately 5% of these intervals will be larger.

b) Nothing can guarantee absolutely that you will get a smaller interval. One can only say the chance of obtaining a smaller interval is 0.05.

c) Use a larger confidence level.

d) Use a smaller confidence level.

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Question 19 (2.85 points)

In formulating hypotheses for a statistical test of significance, the null hypothesis is often

Question 19 options:

a) 0.05.

b) a statement that the data are all 0.

c) the probability of observing the data you actually obtained.

d) a statement of “no effect” or “no difference.”

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Question 20 (2.85 points)

In their advertisements, the manufacturers of a certain brand of breakfast cereal would like to claim that eating their oatmeal for breakfast daily will produce a mean decrease in cholesterol of more than 10 points in one month for people with cholesterol levels over 200. In order to determine if this is a valid claim, they hire an independent testing agency, which then selects 25 people with a cholesterol level over 200 to eat their cereal for breakfast daily for a month. The agency should be testing the null hypothesis H0: μ = 10 and the alternative hypothesis

Question 20 options:

a) Ha: μ 10 ± .

b) Ha: μ < 10.

c) Ha: μ >10.

d) Ha: μ 10.

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Question 21 (2.85 points)

In a test of statistical hypotheses, the P-value tells us

Question 21 options:

a) if the alternative hypothesis is true.

b) the smallest level of significance at which the null hypothesis can be rejected.

c) the largest level of significance at which the null hypothesis can be rejected.

d) if the null hypothesis is true.

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Question 22 (2.85 points)

The scores of a certain population on the Wechsler Intelligence Scale for Children (WISC) are thought to be Normally distributed, with mean μ and standard deviation σ = 10. I wish to test whether the mean for this population differs from the national average of 100, so I use the hypotheses H0: μ = 100 and Ha: μ ? 100, based on an SRS of size 25 from the population. I calculate a 95% confidence interval for μ and find it to be 100.76 to 106.24. Which of the following is true?

Question 22 options:

a) I would reject H0 at level 0.05.

b) I would reject Ha at level 0.05.

c) The P-value is 0.05.

d) A mistake has almost certainly been made. The confidence interval must contain μ = 100 at least 95% of the time.

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Question 23 (2.85 points)

Suppose we are testing the null hypothesis H0: μ = 20 and the alternative Ha: μ 20, for a normal population with σ = 5. A random sample of 25 observations are drawn from the population, and we find the sample mean of these observations is = 17.6. The P-value is closest to

Question 23 options:

a) 0.1336.

b) 0.0668.

c) 0.0082.

d) 0.0164.

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Question 24 (2.85 points)

The mean area μ of the several thousand apartments in a new development by a certain builder is advertised to be 1100 square feet. A tenant group thinks this is inaccurate, and suspects that the actual average area is less than 1100 square feet. In order to investigate this suspicion, the group hires an engineer to measure a sample of apartments to verify its suspicion. The appropriate null and alternative hypotheses, H0 and Ha, for μ are

Question 24 options:

a) H0: μ = 1100 and Ha: μ > 1100.

b) H0: μ = 1100 and Ha: μ 1100.

c) The hypotheses cannot be specified without knowing the size of the sample used by the engineer.

d) H0: μ = 1100 and Ha: μ < 1100.

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Question 25 (2.85 points)

In a statistical test of hypotheses, we say the data are statistically significant at level α if

Question 25 options:

a) the P-value is larger than α.

b) α is small.

c) the P-value is less than α.

d) α = 0.05.

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Question 26 (2.85 points)

A researcher wishes to determine if the use of an herbal extract improves memory. Subjects will take the herbal extract regularly during a 10-week period of time. After this course of treatment, each subject has his or her memory tested using a standard memory test. Suppose the scores on this test of memory for all potential subjects taking the herbal extract follow a Normal distribution with mean μ and standard deviation σ = 6. Suppose also, that in the general population of all people, scores on the memory test follow a Normal distribution, with mean 50 and standard deviation σ = 4. The researcher, therefore, decides to test the hypotheses

H0: μ = 50, Ha: μ > 50

To do so, the researcher has 100 subjects follow the protocol described above. The mean score for these students is = 55.2 and the P-value is less than 0.0001.

Reference: Ref 16-1

Suppose that another researcher attempts to replicate the study described above. She uses a sample of 10 subjects and observes a sample mean memory score of 55.2, the same as the sample mean described in the first study. It is appropriate to conclude which of the following?

Question 26 options:

a) She has obtained the same sample mean as the first researcher, so her P-value will be the same as that of the first researcher.

b) She has obtained the same sample mean as the first researcher, but her P-value will be smaller than that of the first researcher.

c) She has obtained the same sample mean as the first researcher, but her P-value will be greater than that of the first researcher.

d) None of the above

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Question 27 (2.85 points)

A radio show conducts a phone-in survey each morning. Listeners are asked to call in with a response to the question of the day. One morning in 2011, listeners were asked if they supported or opposed term limits for members of Congress. Remarkably, 88% of listeners that called in favored term limits. We may safely conclude that

Question 27 options:

a) nothing, except that a great majority of those with strong enough feelings on the issue to call in are in favor of congressional term limits. We cannot generalize any of this survey's results to any larger population.

b) there is strong evidence that the majority of Americans believe that there should be congressional term limits.

c) there is overwhelming approval for congressional term limits among Americans generally.

d) it is unlikely that if all Americans were asked their opinion, the results would differ from that obtained in the poll.

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Question 28 (2.85 points)

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels over 200 is Normally distributed, with mean (in milligrams) 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 over 200, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this, you test the hypotheses

H0: μ = 10, Ha: μ < 10

and you obtain a P-value of 0.052. Which of the following is true?

Question 28 options:

a) You have failed to obtain any evidence for Ha.

b) There is some evidence against H0, and a study using a larger sample size may be worthwhile.

c) At the α = 0.05 significance level, you have proved that H0 is true.

d) This should be viewed as a pilot study, and the data suggest that further investigation of the hypotheses will not be fruitful at the α = 0.05 significance level.

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Question 29 (2.85 points)

You plan to construct a confidence interval for the mean μ of a Normal population with (known) standard deviation σ. Which of the following will reduce the size of the margin of error?

Question 29 options:

a) Use a lower level of confidence.

b) Increase the sample size.

c) Reduce σ.

d) All of the above

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Question 30 (2.85 points)

The scores of a certain population on the Wechsler Intelligence Scale for Children (WISC) are thought to be Normally distributed with mean μ and standard deviation σ = 10. A simple random sample of 25 children from this population is taken and each is given the WISC. The mean of the 25 scores is = 104.32.

Reference: Ref 14-2

Suppose a histogram of the 25 WISC scores is the following.

Based on this histogram, we would conclude that

Question 30 options:

a) the 95% confidence interval given in the previous question is very reliable.

b) the 95% confidence interval given in the previous question is not very reliable.

c) the 95% confidence interval given in the previous question is actually a 90% confidence interval.

d) the 95% confidence interval given in the previous question is actually a 99% confidence interval.

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Question 31 (2.85 points)

You measure the lifetime of a random sample of 64 tires of a certain brand. The sample mean is = 50 months. Suppose that the lifetimes for tires of this brand follow a normal distribution, with unknown mean μ and standard deviation σ = 5 kg. A 99% confidence interval for μ is

Question 31 options:

a) 48.78 to 51.22.

b) 40.2 to 59.8.

c) 49.80 to 50.20.

d) 48.39 to 51.61.

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Question 32 (2.85 points)

A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean grade point average (GPA) for male and female scholarship athletes (P = 0.287). This means that

Question 32 options:

a) the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample if there is no difference in mean GPAs is 0.287.

b) the maximum difference in GPAs between male and female scholarship athletes is 0.287.

c) the GPAs for male and female scholarship athletes are identical, except for 28.7% of the athletes.

d) the chance that a pair of randomly chosen male and female scholarship athletes would have a significant difference in GPAs is 0.287.

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Question 33 (2.85 points)

Suppose that two very large companies (A and B) each select random samples of their employees. Company A had 5000 employees. Company B has 15,000 employees. In both surveys, the company will record the number of sick days taken by each sampled employee.

Reference: Ref 11-2

If each company randomly selects 50 employees for the survey, which of the following is true about the sampling distributions of the sample means (the mean number of sick days)?

Question 33 options:

a) The sampling distributions of the sample means will have about the same standard deviation. The standard deviation for a sampling distribution of a sample mean depends only on the sample size, not the population (company) size.

b) Since Company A is surveying a higher percent of its employees, the standard deviation of the sampling distribution for its sample mean will be smaller than that for Company B (the larger company). Larger companies should take larger samples.

c) Since Company B is a larger company, the standard deviation for its sampling distribution of the sample mean is smaller. The larger a population, the smaller the standard deviation of a sample mean's sampling distribution.

d) None of the above

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Question 34 (2.85 points)

The sampling distribution of a statistic is

Question 34 options:

a) the probability that we obtain the statistic in repeated random samples.

b) the mechanism that determines whether randomization was effective.

c) the extent to which the sample results differ systematically from the truth.

d) the distribution of values taken by a statistic in all possible samples of the same size from the same population.

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Question 35 (3.1 points)

In testing hypotheses, if the consequences of failing to reject a null hypothesis that is actually false are very serious, we should

Question 35 options:

a) use a very large level of significance.

b) insist that the level of significance be smaller than the P-value.

c) insist that the P-value be smaller than the level of significance.

d) use a very small level of significance.

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