MAT 155 – Lab 11



MAT 155 – Lab

Confidence Intervals for the Population Mean – TI83, TI84

I. Estimate μ with σ unknown

Example: A random sample of 16 wolf dens showed the number of pups in each den. The data below represents this. Find a 90% confidence interval for the population mean number of pups in such dens.

|5 |8 |7 |5 |3 |4 |3 |9 |

|5 |8 |5 |6 |5 |6 |4 |7 |

To start this problem – You must figure out the given information using the raw data. In this example, you can find that n = 16 by counting the number of data values or reading the problem. The calculator will calculate the other sample statistics as it creates the confidence interval. You must assume that the date come from a normal population since the sample is so small. Use the following steps as a guide to find the confidence interval. [pic]

1. Enter the data in list L1. Press STAT. Under the EDIT menu select 1:Edit by pressing ENTER.

2. Since σ is unknown, we use the t-distribution to calculate the confidence interval. Press STAT and scroll right to TESTS. Select 8:TInterval.

3. Beside the Input line, highlight the word Data and press ENTER. In this example, we have raw data that we have put into the list.

4. Beside List make sure it has L1 or wherever you have entered your data and there should be a 1 beside Freq.

5. The confidence level in this problem is 90%. So beside C-Level type .90.

6. Press ENTER on the word Calculate to find the confidence interval.

7. You should have the following output:

8. The confidence interval would be written like: 4.84 < μ < 6.4

9. The 90% confidence interval for the population mean number of pups is 4.84 pups to 6.41 pups. The calculator also states the sample mean and standard deviation of the given data.

II. Applications for Confidence Intervals for a Population Mean

1. A recent Gallup poll asked 1006 Americans, “During the past year, about how many books, either hardcover or paperback, did you read either all or part of the way through?” Results of the survey indicated that the sample mean was 13.4 books and the sample standard deviation 16.6 books. Construct a 99% confidence interval for the mean number of books that Americans read either all or part of during the preceding year. Interpret the interval.

2. A random sample of 51 fatal crashes in 2009 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 grams per deciliter (g/dL) with a standard deviation of 0.010 g/dL.

a. A histogram of blood alcohol concentrations in fatal accidents shows that BACs are highly skewed right. Explain why a large sample size is needed to construct a confidence interval for the mean BAC of fatal crashes with a positive BAC.

b. Determine and interpret a 90% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC.

c. All 50 states and the District of Columbia use a BAC of 0.08 g/dL as the legal intoxication level. Is it possible that the mean BAC of all drivers involved in fatal accidents who are found to have positive BAC values is less than the legal intoxication level? Explain.

3. After scanning the UPC code for a 4 GB flash memory card using his smartphone, Michael Sullivan received the following prices for the goods at a variety of online retailers. Treat these data as a simple random sample of all online retailers.

|12.25 |13.49 |13.76 |14.75 |14.99 |

|14.99 |15.27 |15.95 |17.88 |20.49 |

|Source: ScanLife |

a. Determine a point estimate for the population mean price of the memory card.

b. Use the normal probability plot and boxplot shown next to assist in verifying the requirements for constructing a confidence interval for the mean price.

[pic]

c. Construct and interpret a 95% confidence interval to estimate the mean price of a 4 GB flash memory card.

d. Construct and interpret a 99% confidence interval to estimate the mean price of a 4 GB flash memory card.

e. What happens to the interval as the level of confidence is increased? Explain why this is a logical result.

III. Estimating p

Example: The public television station BPBS wants to find the percent of its viewing population who give donations to the station. 300 randomly selected viewers were surveyed, and it was found that 123 made contributions to the station. Find a 95% confidence interval for the probability that a viewer of BPBS selected at random contributes to the station.

To start this problem – you must know that you are finding a confidence interval for a proportion. (One way to tell this by elimination is that there is no mean or standard deviation given). The following steps will guide you through how to calculate this on your calculator. [pic]

1. Press STAT and scroll right to TESTS.

2. Since we are given n and the number of successes out of n trials, we use the standard normal distribution for proportions or Select A:1-PropZInt.

3. The letter x is used to count the number of successes (the letter r is used in the text). Enter 123 for x and 300 for n.

4. The confidence level in this problem is 95%. So beside C-Level type .95.

5. Press ENTER on the word Calculate to find the confidence interval.

6. You should have the following output:

7. The confidence interval would be written like the following: .35 < p < .47

8. The 95% confidence interval for the probability that a viewer of BPBS selected at random contributes to the station is between about 35% and 47% of the time.

9. Notice that p-hat is .41 because 123/300 = .41 in the given sample information.

IV. Applications for Confidence Intervals about a population proportion, p

1. Many types of error will cause a computer program to terminate or give incorrect results. One type of error is punctuation. For instance, if a comma is inserted in the wrong place, the program might not run. A study of programs written by students in an introductory programming course showed that 75 out of 300 errors selected at random were punctuation errors. Find a 99% confidence interval for the proportion of errors made by beginning programming students that are punctuation errors.

2. In a Gallup poll conducted December 11-14, 2003, 455 of 1011 randomly selected adults aged 18 and older said they had too little time for relaxing or doing nothing.

a. Verify that the requirements for constructing a confidence interval about [pic]are satisfied.

b. Construct a 92% confidence interval for the proportion of adults aged 18 and older who say they have too little time for relaxing or doing nothing. Interpret this interval.

c. Construct a 96% confidence interval for the proportion of adults aged 18 and older who

d. What is the effect of increasing the level of confidence on the width of the interval?

Funny video to watch before completing the problem below about washing hands in public restrooms: (Type this link into a new internet browser).

3. The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies during 2007, both of which were conducted by Harris Interactive. In one of the studies, 1,001 adults were interviewed by telephone and asked about their hand-washing habits. In the other study, the hand-washing behavior of 6,067 adults was inconspicuously observed within public rest rooms in four U.S. cities.

a. In the telephone interviews, 921 of the adults said they always wash their hands in public rest rooms. Use this result to obtain a point estimate for the proportion of adults who say they always wash their hands in public rest rooms.

b. Verify that the requirements for constructing a confidence interval for p from part (a) are satisfied.

c. Using the results from the telephone interviews, construct a 95% confidence interval for the proportion of adults who say they always wash their hands in public rest rooms.

d. In the observational study, 4,679 of the 6,076 adults were observed washing their hands. Use this result to obtain a point estimate for the proportion of adults who wash their hands in public rest rooms.

e. Assuming that the requirements for constructing a confidence interval for p from part (d) are satisfied, use the results from the observational study, construct a 95% confidence interval for the proportion of adults who wash their hands in public rest rooms.

f. Based on your findings in parts (a) through (e), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public rest rooms?

V. Submit through the assignment link on Moodle.

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