Sampling Distribution of the Sample Mean — Unit test Answers

The governor of Florida would like to estimate the mean age of all Florida residents. To do so, he randomly selects 500 Florida residents and determines their mean age to be 58.2 years. However, according the most recent census data, the mean age of all Florida residents is 55.6 years. Which of the following properly describes the 500 Florida residents who were randomly selected?

A real estate agent in a large city claims to sell 15% of their houses within two months of the original listing. A random sample of 75 listings is chosen, and 9 were sold within two months of the original listing. Let = the proportion of houses in the random sample that were sold within two months.The probability of 12% or fewer of this real estate agent’s listings being sold within two months is 0.295. Does this result provide convincing evidence against the agent’s claim?

In an urban area, 31% of dog owners pay for someone to walk their dogs. Morgan asks a random sample of 50 dog owners in this urban area if they pay for someone to walk their dog. What is the probability that more than 30% of the owners in the sample pay for someone to walk their dog?Find the z-table here.

The governor of Florida would like to estimate the mean age of all Florida residents. To do so, he randomly selects 500 Florida residents and determines their mean age to be 58.2 years. However, according the most recent census data, the mean age of all Florida residents is 55.6 years. Which of the following properly describes the number 58.2?

Alex’s times for running a mile are Normally distributed with a mean time of 5.28 minutes and a standard deviation of 0.38 seconds. Chris’s times for running a mile are Normally distributed with a mean time of 5.45 seconds and a standard deviation of 0.2 seconds. Ten of Alex’s times and 15 of Chris’s times are randomly selected. Let represent the difference in the mean times for Alex and Chris. Which of the following represents the standard deviation of the sampling distribution for ?

A bottled water company bottles varying sizes of water, from 8-ounce to 1-gallon containers. The company has determined that the mean quantity in their 20-ounce bottles is 20.8 ounces with a standard deviation of 0.6 ounces. The bottling plant manager believes his machines are overfilling the bottles. A random sample of 30 bottles is taken, and the mean number of ounces of water is determined to be 20.9. Under the assumption that the true mean ounces of water is 20.8, 100 simulated means for samples of size 30 are shown in the dotplot.Using the dotplot, is there evidence that machines are overfilling the 20-ounce bottles?

A student decides to spin a dime and determine the proportion of times it lands on heads. The student spins the dime 25 times and records that it lands on heads 17 times. Let p = the true proportion of times the dime would land on heads when spun. Under the assumption that the true proportion is 0.5, 100 simulated proportions for samples of size 25 are shown in the dotplot. Using the dotplot, what do the two dots above 0.70 represent?

The proportion of all high school students who watch national news is p = 0.47. A random sample of 50 high school students is selected. Which of the following is the mean of the sampling distribution of ?

The daily temperatures for the winter months in Virginia are Normally distributed with a mean of 59°F and a standard deviation of 10°F. A random sample of 10 temperatures is taken from the winter months and the mean temperature is recorded. What is the standard deviation of the sampling distribution of the sample mean for all possible random samples of size 10 from this population?

The number of turns of a pencil sharpener needed to sharpen a brand W pencil is approximately Normally distributed with a mean of 4.6 and a standard deviation of 0.67. The number of turns needed to sharpen a brand H pencil is approximately Normally distributed with a mean of 5.2 and a standard deviation of 0.33. If 30 pencils of each brand are randomly selected and sharpened, what is the probability that the brand W pencils will have a higher mean number of turns needed to sharpen than brand H?

The proportion of all high school students who watch national news is p = 0.57. A random sample of 60 high school students is selected. Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

The distribution of tips given by customers who buy one cup of coffee is bimodal with a mean of $0.29 and a standard deviation of $0.1164. If a random sample of 50 tips from customers who buy one cup of coffee is selected, is it appropriate to calculate the probability of a sample mean being more than $0.32 using an approximately Normal model?

The number of pieces of dog food in a one-cup scoop is approximately Normally distributed with a mean of 205 pieces and a standard deviation of 9.2 pieces. If a random sample of 15 scoops of dog food is selected, what is the probability that the mean number of pieces of food will be less than 200 pieces?

The ages of the 5 officers for a school club are 18, 18, 17, 16, and 15. The proportion of officers who are younger than 18 is 0.6. The table displays all possible samples of size 2 and the corresponding proportion for each sample.Using the proportions in the table, is the sample proportion an unbiased estimator?

Among the senior class at a high school, 55% of Ms. Keating’s students plan on majoring in a branch of STEM, while 49% of Ms. Lewis’s students plan on majoring in a branch of STEM. Suppose Ms. Keating chooses 25 of her students at random and Ms. Lewis chooses 23 of her students at random. Since nKpK, nK (1 – pK) and nLpL, nL (1 – pL) are all greater than 10, the Normal condition is met. Let K = the proportion of Ms. Keating’s students from the sample who plan on majoring in a branch of STEM, and let L = the proportion of Ms. Lewis’s students from the sample who plan on majoring in a branch of STEM. What is the probability that the proportion of students who plan on majoring in a branch of STEM is greater for Ms. Keating?Find the z-table here.

The owner of a tire company wants to determine how long a new type of tire tread will last by measuring the length of time, in days, it takes for the tire tread to reach 2 mm. Let μ = the true mean number of days it takes for the tire tread to wear to 2 mm. Which of the following sample sizes will have the least variability in the sampling distribution of the sample mean?

In a large high school, 37% of the teachers believe that five minutes is not enough time for students to change classes. However, 89% of the students believe that five minutes is not enough time for students to change classes. Let and be the sample proportions of teachers and students, respectively, who believe that five minutes is not enough time for students to change classes. Suppose 28 teachers and 100 students are selected at random and asked their opinion on the amount of time students have to change class.Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of ?

A spinner with four equal sectors labeled "purple,” "blue,” "orange,” and "green” is spun 45 times. What is the probability that the spinner will land on "purple” in fewer than 20% of the spins?Find the z-table here.

A statistics teacher has 20 students in her class. Here is a graph showing the scores of the 20 students on a recent test.The teacher selects a random sample of 5 students from the class. Here is a graph displaying the scores of the students she selected.The teacher calculates the mean of this sample of 5 scores and plots it on a number line. She then repeats this process 100 times. Here are the results.

The salaries of professional baseball players are heavily skewed right with a mean of $3.2 million and a standard deviation of $2 million. A baseball analyst randomly selects 40 athletes and records the mean salary. What is the shape of the distribution of the sample mean for all possible random samples of size 40 from this population?