Exam 15: Sampling Distribution Models

In a large statistics class,the professor has each student toss a coin 58 times and calculate the proportion of his or her tosses that were tails.The students then report their results,and the professor plots a histogram of these several proportions.Should a Normal model be used here?

The weight of crackers in a box is stated to be 450 grams.The amount that the packaging machine puts in the boxes is believed to have a Normal model with mean 465 grams and standard deviation 30 grams.What is the probability that the mean weight of a $50 - \mathrm { box }$ case of crackers is above 450 grams?

The weight of crackers in a box is stated to be 450 grams.The amount that the packaging machine puts in the boxes is believed to have a Normal model with mean 465 grams and standard deviation 30 grams.What is the probability that the mean weight of a $50 - \mathrm { box }$ case of crackers is below 450 grams?

In a large statistics class,the professor has each student toss a coin 14 times and calculate the proportion of his or her tosses that were tails.The students then report their results,and the professor plots a histogram of these several proportions.Should a Normal model be used here?

A sample is chosen randomly from a population that was strongly skewed to the right.Describe the sampling distribution model for the sample mean if the sample size is small.

In a large class,the professor has each person toss a coin 200 times and calculate the proportion of his or her tosses that were tails.The students then report their results,and the professor records the proportions.One student claims to have tossed her coin 200 times and found 58% tails.What do you think of this claim? Explain your response.

Consider simulating the average roll of five 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 five rolls is between (and including)15 and 20.

Consider simulating the average roll of two 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 two rolls is between (and including)6 and 8.

Based on past experience,a bank believes that 5% of the people who receive loans will not make payments on time.The bank has recently approved 300 loans.What is the probability that over 6% of these clients will not make timely payments?

If the students toss the coin 50 times each,about 68% should have proportions between what two numbers?

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 at least $12?

A candy company claims that 20% of the jelly beans in its spring mix are pink.Suppose that the candies are packaged at random in small bags containing about 200 jelly beans.A class of students opens several bags,counts the various colors of jelly beans,and calculates the proportion that are pink.Is it appropriate to use a Normal model to describe the distribution of the proportion of pink jelly beans?

The amount filled by a certain bottling machine,X,is approximately normally distributed with unknown mean,μ,and standard deviation,σ =1.If n=16 bottles were randomly sampled,what is the probability that the mean fill in the sample is within 0.3 ounces of the population mean μ?

A candy company claims that 17% of the jelly beans in its spring mix are pink.Suppose that the candies are packaged at random in bags containing about 400 jelly beans.A class of students opens several bags,counts the various colours of jelly beans,and calculates the proportion that are pink.In one bag,the students found 14% of the jelly beans were pink.Is this an unusually small proportion of pink jelly beans? Explain your response.

A 1000-acre farm historically averages 193 bushels per acre with a standard deviation of 16 bushels per acre.Fifty acres are sampled and the mean yield determined.If we imagined all the possible random samples of 50 acres we could take and looked at all the sample means,is it appropriate to assume this data will be well modeled by a Normal distribution?

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 12% blue jelly beans?

If we imagine all the possible random samples of 250 students at this university,68% of the samples should have means between what two numbers?

When a truckload of oranges arrives at a packing plant,a random sample of 125 is selected and examined.The whole truckload will be rejected if more than 8% of the sample is unsatisfactory.Suppose that in fact 10% of the oranges on the truck do not meet the desired standard.What is the probability that the shipment will be rejected?

When a truckload of oranges arrives at a packing plant,a random sample of 125 is selected and examined.The whole truckload will be rejected if more than 8% of the sample is unsatisfactory.Suppose that in fact 11% of the oranges on the truck do not meet the desired standard.What is the probability that the shipment will be accepted anyway?

Consider simulating the average roll of three 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 three rolls is between (and including)9 and 12.