Unit 5

In a large city, 46% of adults support the local football team building a new stadium.

If a poll is taken from a random sample of 80 adults in the large city, which of the following properly describes the sampling distribution of the sample proportion of adults who support the stadium?

• (A) μp​=36.8, σp=4.46, the distribution is approximately normal.

• (B) μp​=36.8, σp=4.46, the shape of the distribution is unknown.

• (C) μp​​=0.46, σp=0.056, the distribution is approximately normal.

• (D) μp​​=0.46, σp=0.056, , the shape of the distribution is unknown.

• (E) μp​​=43.2, σp​​=4.46, the distribution is binomial.

Answer explanation

The shape is approximately normal since the expected number of successes = 36.8 and the expected number of failures = 43.2 are both greater than 10.

Which of the following statements is true?

• (A) A parameter is a number that describes some characteristic of a sample.

• (B) An unbiased estimator is any statistic that is taken from a sample chosen by random methods.

• (C) A sampling distribution is the distribution of a statistic calculated from all possible samples of the same size from the same population.

• (D) The variability of a population distribution will decrease as the sample size increases.

• (E) A normal approximation can always be used for the sampling distribution of p^​ as long as the sample size is greater than 30.

The weight of a single bag checked by an airplane passenger follows a distribution that is right skewed with a mean of 38 pounds and a standard deviation of 6.2 pounds.

If a random sample of 96 bags is selected, what is the probability that the average weight of the bags exceeds 40 pounds?

• (A) 0.0008

• (B) 0.0011

• (C) 0.3735

• (D) 0.9992

• (E) It is not appropriate to use a normal distribution to calculate probability in this situation.

Answer explanation

A normal distribution should be used for the calculation. The z-score for a bag exceeding 40 pounds is 3.16 and P(z>3.16) = 0.0008.

Below are histograms displaying the values taken by three sample statistics in several hundred samples from the same population. Each histogram has the same scale and the true value of the population parameter is marked by an arrow on each histogram.

Which statistic is the best estimator of the parameter?

• (A) Statistic A because A and B are unbiased but A has higher variability.

• (B) Statistic B because A and B are unbiased but B has lower variability.

• (C) Statistic C because it is always better to underestimate rather than overestimate.

• (D) Statistics A or B because the population parameter is in the center of the sampling distributions for both.

• (E) Statistics A, B or C because all contain the population parameter in the sampling distributions.

Answer explanation

The best statistic is unbiased, with the lowest variability. 

A company intends to collect a random sample of size n from a population with population proportion of interest p.

Which of the following situations would result in the smallest standard deviation of the sampling distribution of p^​?

• (A) The population proportion p=0.5, and the sample size is n.

• (B) The population proportion p≠0.5, and the sample size is n.

• (C) The population proportion p=0.5, and the sample size is 2n.

• (D) The population proportion p≠0.5, and the sample size is 2n.

• (E) There is not enough information provided to determine the smallest standard deviation of the sampling distribution.

Answer explanation

Any population proportion not equal to 0.5 will decrease the product of p(1−p). Also, increasing the sample size will decrease the variability of the sampling distribution.

Kent county and Oakland county are located on the west and east sides of Michigan, respectively. Information about the yearly household income is given in the table below. Both distributions are strongly skewed right. A random sample of 100 households is taken from each of the very large populations.

Which of the following describe the sampling distribution of the difference in sample means?

• (A) μxˉO​−xˉK​​=12060, σxˉO​−xˉK​​=870, the distribution is approximately normal.

• (B) μxˉO​−xˉK​​=12060, σxˉO​−xˉK​​=6259.91, the distribution is strongly skewed right.

• (C) μxˉO​−xˉK​​=12060, σxˉO​−xˉK​​=6259.91, the distribution is approximately normal.

• (D) μxˉO​−xˉK​​=12060, σxˉO​−xˉK​​=8810, the distribution is strongly skewed right.

• (E) μxˉO​−xˉK​​=12060, σxˉO​−xˉK​​=8810, the distribution is approximately normal.

Answer explanation

A recent report states that 89% of Americans consider themselves above average drivers. A local newspaper is planning on conducting a survey to investigate whether this is true locally.

If the newspaper assumes that the 89% claim is true and plans to use the normal approximation to calculate probabilities associated with their sample proportion, which sample size would be most appropriate?

• (A) 10

• (B) 11

• (C) 30

• (D) 91

• (E) The number depends on the size of the population.

Answer explanation

The requirements for the normal condition np≥10 and n(1−p)≥10 are only satisfied with sample sizes of 91 or greater.

Daren and Josh are pretty good free throw shooters. Daren makes 75% of the free throws he attempts. Josh makes 80% of his free throws.

Suppose we take separate random samples of 50 free throws each from Daren and Josh, and record the proportion of free throws that are made by each. Which of the following best describes the sampling distribution of p^​D​−p^​J​?

• (A) Strong skew, with mean -0.05 and standard deviation 0.083

• (B) Approximately normal, with mean -0.05 and standard deviation 0.083

• (C) Shape cannot be determined, with mean -0.05 and standard deviation 0.083

• (D) Strong skew, with mean -0.05 and standard deviation 0.118

• (E) Approximately normal, with mean -0.05 and standard deviation 0.118

Answer explanation

Donner Summit, California, is a popular ski resort area. Over the past 60 years, the annual snowfall totals of Donner Summit have followed a distribution that is strongly skewed right with a mean of 404 inches and a standard deviation of 129 inches.

If many samples of size 9 were taken, which of the following would best describe the shape of the sampling distribution of xˉ?

• (A) The shape is approximately normal since the sample size is reasonably large.

• (B) The shape is skewed right since np≥10 and n(1−p)≥10 has not been met.

• (C) The shape is equally as skewed right as the population distribution.

• (D) The shape is skewed right but less so than the population distribution.

• (E) Cannot be determined from the given information.

Answer explanation

The small sample size means that the shape of the sampling distribution will be similar to the shape of the population distribution, but the central limit theorem states that the sampling distribution will tend towards normal as the sample size increases.

The heights of all adult males in Croatia are approximately normally distributed with a mean of 180 cm and a standard deviation of 7 cm. The heights of all adult females in Croatia are approximately normally distributed with a mean of 158 cm and a standard deviation of 9 cm.

If independent random samples of 10 adult males and 10 adult females are taken, what is the probability that the difference in sample means (males – females) is greater than 20 cm?

• (A) 0.3463

• (B) 0.6537

• (C) 0.6827

• (D) 0.7104

• (E) 0.8687

Answer explanation

The sampling distribution of xˉ1​−xˉ2​ has a mean of 22 and standard deviation of 3.6055. P(z > -0.5547) = 0.7104.