CASE STUDY CH.6

DROPBOX ASSIGNMENT

A spice manufacturer has a machine that fills bottles. The bottles are labeled 16 grams net weight so the company wants to have that much spice in each bottle. The company knows that just like any packaging process this packaging process is not perfect and that there will some variation in the amount filled. If the machine is set at exactly 16 grams and the normal distribution applies, then about half of the bottles will be underweight making the company vulnerable to bad publicity and potential lawsuits. To prevent underweight bottles, the manufacturer has set the mean a little higher than 16 grams. Based on their experience with the packaging machine, the company believes that the amount of spice in the bottle fits a normal distribution with a standard deviation of 0.2 grams. The company decides to set the machine to put an average 16.3 grams of spice in each bottle. Based on the above information answer the following questions:

1) What percentage of the bottles will be underweight? (5 Points)

2) The company's lawyers says that the answer obtained in question 1 is too high. They insist that no more then 4% of the bottles can be underweight and the company needs to put a little more spice in each bottle. What mean setting do they need? (5 Points)

3) The company CEO says that they do not want to give away too much free spice. She insists that the machine be set no higher than 16.2 grams (for the average) and still have only 4% underweight bottles as specified by the lawyers. This can be only accomplished by reducing the standard deviation. What standard deviation must the company achieve to meet the mandate from the CEO? (4 Points)

4) A disgruntled employee decides to set the machine to put an average 17.4 grams of spice in each bottle. What % of the bottles will be over weight (use standard deviation of 0.2 grams for this question)? (5 Points Hint: this question is similar to Question 1 but make sure you draw a diagram so as to answer this question correctly)

5) Can you think of a practical way as to how the company can reduce the standard deviation for this bottle filling process? (1 Point)

The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 80​% confident that his estimate is within seven percentage points of the true population​ percentage? Complete parts​ (a) through​ (c) below

​a) Assume that nothing is known about the percentage of adults who have heard of the brand.

n=

(Round up to the nearest​ integer.)

​b) Assume that a recent survey suggests that about 86​% of adults have heard of the brand.

n=

​(Round up to the nearest​ integer.)

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,300. We randomly survey ten teachers from that district. (Round your answers to the nearest dollar.)

(a) Find the 90^th percentile for an individual teacher's salary.

(b) Find the 90^th percentile for the average teacher's salary.

A typical adult has an average IQ score of 105 with a standard deviation of 20. If 19 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 85 and 123 points? (Round your answer to five decimal places.)

Define 2 different measures of correlation of 2 data sets to each other.

List all basic distributions for which:

a) MLE is unbiased, but Method of Moments (MM) estimator is biased

b) MLE is biased, but MM estimator is unbiased

An industrial plant discharges water into a river. An environmental protection agency has studied the discharged water and found the lead concentration in the water (in micrograms per litre) has a normal distribution with population standard deviation σ = 0.7 μg/l. The industrial plant claims that the mean value of the lead concentration is 2.0 μg/l. However, the environmental protection agency took 10 water samples and found that the mean is 2.56 μg/l. A hypothesis test is carried out to determine whether the lead concentration population mean is higher than the industrial plant claims. (Use 1% level of significance). An appropriate test for this one population hypothesis problem is to use the _______.

A random sample of 20 observations results in 11 successes. [You may find it useful to reference the z table.]

a. Construct the an 90% confidence interval for the population proportion of successes. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)  

Construct the an 90% confidence interval for the population proportion of failures. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)

Many consumer groups feel that the U.S. Food and Drug Administration (FDA) drug approval process is too easy and, as a result, too many drugs are approved that are later found to be pushed for a more lenient approval process so that pharmaceutical companies can get new drugs approved more easily and quickly. Consider a null hypothesis that a new, unapproved drug is unsafe and an alternative hypothesis that a new, unapproved drug is safe.

1. Differentiate Type 1 and Type 2 errors.
2. Given the case scenario, explain the risks of committing a Type 1 or Type 2 error.
3. Which type of error are consumer groups trying to avoid?
4. Which type of error are industry lobbyists trying to avoid?

One state lottery game has contestants select 5 different numbers from 1 to 45. The prize, if all numbers are matched is 2 million dollars. The tickets are $2 each.

1) How many different ticket possibilities are there?

2) If a person purchases one ticket, what is the probability of winning? What is the probability of losing?

3) Occasionally, you will hear of a group of people going in together to purchase a large amount of tickets. Suppose a group of 30 purchases 6,000 tickets. a) How much would each person have to contribute? b) What is the probability of the group winning? Losing?

4) How much would it cost to “buy the lottery”, that is, buy a ticket to cover every possibility? Is it worth it?

5) Create a probability distribution table for the random variable x = the amount won/lost when purchasing one ticket.

6) In fair games, the expected value will be $0. This means that if the game is played many…many times, then one is expected to break even eventually. This is never true for Casino and Lottery games. Find the expected value of x = the amount won/lost when purchasing one ticket.

7) Interpret the expected value. See section 4.2 in the textbook for an example on how to interpret the expected value.

8) Fill in the following table using the expected value.

Please answer all questions! I will rate you!

QUESTION 5

From a sample of 500 college students it was found that 300 of them had taken a statistics course.

Construct a 95% confidence interval for the proportion of college students who have taken a statistics course. What is the LOWER BOUND on the interval? Round your answer to three decimal places (i.e. 0.123).

4 points   

QUESTION 6

From a sample of 500 college students it was found that 300 of them had taken a statistics course.

Construct a 95% confidence interval for the proportion of college students who have taken a statistics course. What is the UPPER BOUND on the interval? Round your answer to three decimal places (i.e. 0.123).

If I toss a fair coin 50,000 times which of the following is true?

a) the number of heads should be between 15,000 and 25,000.

b) the proportion of heads should be close to 50%.

c) the proportion of heads in these tosses is a parameter.

d) the number of heads should be exactly 25,000.

e) the proportion of heads will be close to 1.

When looking at statistics in criminal justice, how do you feel the mean, median, and mode are useful? Do you feel that there are times when one is more valuable than another? How can these be used to help deter crime?

Suppose you are working for a regional residential natural gas utility. For a sample of 95 customer visits, the staff time per reported gas leak has a mean of 219 minutes and standard deviation 34 minutes. The VP of network maintenance hypothesizes that the average staff time devoted to reported gas leaks is 226 minutes. At a 5 percent level of significance, what is the upper bound of the interval for determining whether to accept or reject the VP's hypothesis? Note that the correct answer will be evaluated based on the z-values in the summary table in the Teaching Materials section. Please round your answer to the nearest tenth.

A particular fruit's weights are normally distributed, with a mean of 718 grams and a standard deviation of 27 grams.

If you pick 12 fruits at random, then 16% of the time, their mean weight will be greater than how many grams?

Give your answer to the nearest gram.

1) You test a new drug to reduce blood pressure. A group of 15 patients with high blood pressure report the following systolic pressures (measured in mm Hg): ̄y s before medication: 187 120 151 143 160 168 181 197 133 128 130 195 130 147 193 157.53 27.409 after medication: 187 118 147 145 158 166 177 196 134 124 133 196 130 146 189 156.40 27.060 change: 0 2 4 -2 2 2 4 1 -1 4 -3 -1 0 1 4 1.133 2.295 a) Calculate a 90% CI for the change in blood pressure. b) Calculate a 99% CI for the change in blood pressure. c) Does either interval (the one you calculated in (a) or (b)) include 0? Why is this important? d) Now conduct a one sample t-test using μ = 0, and α = .10. Are the results consistent with (a)? e) Finally, conduct a one sample t-test using μ = 0, and α = .01. Are the results consistent with (b)? PLEASE TELL HOW TO GET ALPHA I USE .05 AND GOT VALUE 2.145