Exam 15: Sampling Distribution Models

A candy company claims that its jelly bean mix contains 15% blue jelly beans.Suppose that the candies are packaged at random in small bags containing about 200 jelly beans.What is the probability that a bag will contain more than 20% blue jelly beans?

Automatic teller machine receipts show that customer withdrawals have a skewed distribution with a mean of $72 and a standard deviation of $15.Let $\overline{ y}$ represent the mean withdrawal for the next 200 customers at the machine.Describe the sampling distribution model of this mean.

Suppose that the national mean weight for a three-year-old girl is 30 pounds with a standard deviation of 5 pounds.A pediatric office takes a random sample of 100 three-year-old girls,records their weights,and finds the mean.Describe the sampling distribution model of this mean.

The number of hours per week that high school students spend on homework is normally distributed,with a mean of 12 hours and a standard deviation of 3 hours.50 students are chosen at random.Let $\overline{ y}$ represent the mean number of hours spent on homework for this group.Find the probability that $\overline{ y}$ is between 10.8 and 13.0.

A national study reported that 76% of high school graduates pursue a college education immediately after graduation.A private high school advertises that 156 of their 196 graduates last year went on to college.Does this school have an unusually high proportion of students going to university?

Some experts estimate that the length of time people live in a house has a mean of 7.3 years and a standard deviation of 2.2 years.Explain why you could estimate the probability that 100 people selected at random had lived in their houses for an average of 9 years or more,but you could not estimate the probability that an individual had done so.

A grower claims that only 13% of the strawberries he grows are unsatisfactory.When a truckload of his strawberries arrives at a farmers' market,one basket containing approximately 105 strawberries is examined for bruised or rotten fruit.Are the conditions satisfied for use of the Normal model?

Statistics from a weather centre indicate that a certain city receives an average of 23 inches of snow each year,with a standard deviation of 3 inches.Assume that a Normal model applies.A student lives in this city for 4 years.Let $\overline{ y}$ represent the mean amount of snow for those 4 years.Describe the sampling distribution model of this sample mean.

Consider simulating the average roll of twenty fair 6-sided dice.Below is a histogram of the simulated average over $10000$ trials. Based on the simulation,approximate the probability that the average roll is between (and including)3 and 4.Note: This is equivalently,the probability that the sum of the twenty rolls is between (and including)60 and 80.

The number of hours per week that high school students spend on computers is normally distributed,with a mean of 5 hours and a standard deviation of 2 hours.70 students are chosen at random.Let $\overline{ y}$ represent the mean number of hours spent on the computer for this group.Find the probability that $\overline{ y}$ is between 5.1 and 5.7.

Based on past experience,a bank believes that 9% of the people who receive loans will not make payments on time.The bank has recently approved 300 loans.Describe the sampling distribution model of the proportion of clients in this group who may not make timely payments.

Assume that 15% of students at a university wear contact lenses.We randomly pick 200 students.What is the probability that more than 16% of this sample wear contact lenses?

A hair stylist believes the distribution of her tips is Normal,with a mean of $10.50 and a standard deviation of $6.25.What is the probability that her 15 clients this weekend will tip an average of less than $13?

Assume that 32% of students at a university wear contact lenses.We randomly pick 300 students.Describe the sampling distribution model of the proportion of students in this group who wear contact lenses.

Some experts claim that the length of time people live in a house can be described by a Normal distribution with a mean of 7.2 years and a standard deviation of 2.4 years.Suppose you calculate the percentage of homeowners who stay in their house for more than 9 years and the probability that the mean length of time for a sample of 100 homeowners is more than 9 years.If the correct model for the length of time people live in a home is in fact skewed to the right,does that change your answer to either calculation? Explain.

If the students toss the coin 50 times each,about 99.7% should have proportions between what two numbers?

Researchers believe that 7% of children have a gene that may be linked to a certain childhood disease.In an effort to track 50 of these children,researchers test 950 newborns for the presence of this gene.What is the probability that they do not find enough subjects for their study?

If we imagine all the possible random samples of 150 students at this university,95% of the samples should have means between what two numbers?

A restaurant's receipts show that the cost of customers' dinners has a skewed distribution with a mean of $54 and a standard deviation of $18.What is the probability that the next 100 customers will spend an average of less than $50 on dinner?

If the students toss the coin 100 times each,about 95% should have proportions between what two numbers?