STAT 5.1-4

Researchers working for a certain airline are investigating the weight of carry-on bags. The researchers will use the mean weight of a random sample of 800 carry-on bags to estimate the mean weight of all carry-on bags for the airline. Which of the following best describes the effect on the bias and the variance of the estimator if the researchers increase the sample size to 1,300 ?

• The bias will decrease and the variance will remain the same.

• The bias will increase and the variance will remain the same.

• The bias will remain the same and the variance will decrease.

• The bias will remain the same and the variance will increase.

• The bias will decrease and the variance will decrease.

A manufacturer of cell phone batteries claims that the average number of recharge cycles for its batteries is 400. A consumer group will obtain a random sample of 100 of the manufacturer’s batteries and will calculate the mean number of recharge cycles. Which of the following statements is justified by the central limit theorem?

• The distribution of the number of recharge cycles for the sample is approximately normal because the population mean of 400 is greater than 30.

• The distribution of the number of recharge cycles for the sample is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the number of recharge cycles for the population is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the sample means of the number of recharge cycles is approximately normal because the sample size of 100 is greater than 30

• The distribution of the sample means of the number of recharge cycles is approximately normal because the population mean of 400 is greater than 30.

The mean and standard deviation of the sample data collected on continuous variable x are −0.25 and 0.03, respectively. The table shows the relative frequencies of the data in the given intervals.

Based on the table, do the data support the use of a normal model to approximate population characteristics?

• Yes, because the sum of the relative frequencies is 1.00.

• Yes, because the distribution of relative frequencies is very close to the empirical rule for normal models.

• No, because the values are negative and normal models are used for positive values.

• No, because the distribution of relative frequencies is very far from the empirical rule for normal models.

• No, because the sample size and the population parameters are not known.

Question

The following graphs show the sampling distributions for two different point estimators, R and W, of the same population parameter.

Which of the following statements is true?

• Both R and W are unbiased.

• Both R and W are biased.

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  R is biased, and W is unbiased.

• R is unbiased, and W is biased.

• The assessment of bias is not possible, because the sampling distributions display too much variability.

In a recent survey, the proportion of adults who indicated mystery as their favorite type of book was 0.325. Two simulations will be conducted for the sampling distribution of a sample proportion from a population with a true proportion of 0.325. Simulation A will consist of 1,500 trials with a sample size of 100. Simulation B will consist of 2,000 trials with a sample size of 50.

Which of the following describes the center and variability of simulation A and simulation B?

• The centers will roughly be equal, and the variabilities will roughly be equal.

• The centers will roughly be equal, and the variability of simulation A will be greater than the variability of simulation B.

• The centers will roughly be equal, and the variability of simulation A will be less than the variability of simulation B.

• The center of simulation A will be greater than the center of simulation B, and the variability of simulation A will roughly be equal to the variability of simulation B.

• The center of simulation A will be less than the center of simulation B, and the variability of simulation A will be greater than the variability of simulation B

A recent survey indicated that the mean time spent on a music streaming service is 210 minutes per week for the population of a certain country. A simulation was conducted to create a sampling distribution of the sample mean for a population with a mean of 210. The following histogram shows the results of the simulation.

Which of the following would be the best reason why the simulation of the sampling distribution is not approximately normal?

• The samples were not selected at random.

• The sample size was not sufficiently large.

• The population distribution was approximately normal.

• The samples were selected without replacement.

• The sample means were less than the population mean.

The continuous random variable N has a normal distribution with mean 7.5 and standard deviation 2.5. For which of the following is the probability equal to 0 ?

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Data will be collected on the following variables. Which variable is most likely to be approximated by a normal model?

• The distribution of the number of books read last week by middle school students, where the right tail of the distribution is longer than the left

• The distribution of life span, in minutes, for batteries of a certain size, where most life spans cluster around the center of the distribution but with some very low and some very high life spans

• The distribution of ages, in years, of the students at a certain college, where most students are between 18 and 22 years old, but ages greater than 22 will probably be more spread out than ages less than 18

• The distribution of the number of birthdays per month for the employees at a certain company, where the number of birthdays in each month is approximately equal

• The distribution of the length of a stay, in days, in a hospital after surgery, where many patients have very short hospital stays, but some stays are quite lengthy and considered high outliers

Clara recorded 50 numerical observations on a certain variable and then calculated the mean <x> and the standard deviation "s" for the observations. To help decide whether a normal model is appropriate, she created the chart.

In Clara’s chart, the letters a,b,c,d,and e represent the number of observations falling in each interval. Which of the following list of counts for a,b,c,d,and e, respectively, is the best indicator that the variable can be modeled with a normal approximation

• 1, 7, 34, 7, 1

• 1, 10, 28, 10, 1

• 2, 8, 30, 8, 2

• 2, 4, 38, 4, 2

• 5, 5, 30, 5, 5

According to government data, 22 percent of children in the United States under the age of 6 years live in households with incomes that are classified at a particular income level. A simple random sample of 300 children in the United States under the age of 6 years was selected for a study of learning in early childhood. If the government data are correct, which of the following best approximates the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level? (Note: z represents a standard normal random variable.)

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