DSS 210 EXAM 2

Which of the following statements are valid null and alternative hypotheses? -------- a. H0: x̅ ≤ 210; HA: x̅ > 210 b. H0: μ = 120; HA: μ ≠ 120 c. H0: p ≤ 0.24; HA: p > 0.24 d. H0: μ < 252; HA: μ > 252

a. Invalid b. Valid c. Valid d. Invalid

Which of the following types of tests may be performed?

Right-tailed, left-tailed, and two-tailed tests

In general, the null and alternative hypotheses are ______.

mutually exclusive

What is zα/2⁢ for a 95% confidence interval of the population mean?

1.96

Many cities around the United States are installing LED streetlights, in part to combat crime by improving visibility after dusk. An urban police department claims that the proportion of crimes committed after dusk will fall below the current level of 0.84 if LED streetlights are installed. Specify the null and alternative hypotheses to test the police department's claim.

H0: p ≥ 0.84 and HA: p < 0.84

How do the tdf and z distributions differ?

The tdf distribution has broader tails (it is flatter around zero).

We draw a random sample of size 36 from a population with standard deviation 3.5. If the sample mean is 27, what is a 95% confidence interval for the population mean?

[25.8567, 28.1433]

Consider the following hypotheses: -- H0: μ ≤ 12.6 HA: μ > 12.6 --- A sample of 25 observations yields a sample mean of 13.4. Assume that the sample is drawn from a normal population with a population standard deviation of 3.2. -------------------- a-1. Find the p-value. -- a-2. What is the conclusion if α = 0.10? -- a-3. Interpret the results at α = 0.10 -- b-1. Calculate the p-value if the above sample mean was based on a sample of 100 observations. -- b-2. What is the conclusion if α = 0.10? -- b-3. Interpret the results at α = 0.10

a-1. p-value ≥ 0.10 -- a-2. Do not reject H0 since the p-value is greater than α -- a-3. We cannot conclude that the population mean is greater than 12.6. -- b-1. p-value < 0.01 -- b-2. Reject H0 since the p-value is smaller than α -- b-3. We conclude that the population mean is greater than 12.6

A study claims that girls and boys do not do equally well on math tests taken from the 2nd to 11th grades (Chicago Tribune, July 25, 2008). Suppose in a representative sample, 344 of 430 girls and 369 of 450 boys score at proficient or advanced levels on a standardized math test. -- Let p1 represent the population proportion of girls and p2 the population proportion of boys. ----------- a. Construct the 95% confidence interval for the difference between the population proportions of girls and boys who score at proficient or advanced levels -- b. Select the appropriate null and alternative hypotheses to test whether the proportion of girls who score at proficient or advanced levels differs from the proportion of boys. -- c. At the 5% significance level, what is the conclusion to the test? Do the results support the study's claim?

a. (-7.20 , 3.20) -- b. H0: p1 − p2 = 0; HA: p1 − p2 ≠ 0 -- c. Do not reject H0; the study's claim is not supported by the sample data

A study by Allstate Insurance Co. finds that 82% of teenagers have used cell phones while driving (The Wall Street Journal, May 5, 2010). Suppose this study was based on a random sample of 50 teen drivers. [You may find it useful to reference the z table.] ----------------- a. Construct the 99% confidence interval for the proportion of all teenagers that have used cell phones while driving -- b. What is the margin of error with 99% confidence?

a. (0.68 , 0.96) -- b. 0.14

A study reports that recent college graduates from New Hampshire face the highest average debt of $31,048 (The Boston Globe, May 27, 2012). A researcher from Connecticut wants to determine how recent undergraduates from that state fare. He collects data on debt from 40 recent undergraduates. A portion of the data is shown in the accompanying table. Assume that the population standard deviation is $5,000. (first few rows...) 24040 23906 19153 23690 26762 32254 , ... ------------- a. Construct the 95% confidence interval for the mean debt of all undergraduates from Connecticut. -- b. Use the 95% confidence interval to determine if the debt of Connecticut undergraduates differs from that of New Hampshire undergraduates.

a. (23,921.16 , 27,020.04) -- b. The debt of Connecticut undergraduates differs from that of New Hampshire undergraduates.

A polygraph (lie detector) is an instrument used to determine if an individual is telling the truth. These tests are considered to be 95% reliable. In other words, if an individual lies, there is a 0.95 probability that the test will detect a lie. Let there also be a 0.005 probability that the test erroneously detects a lie even when the individual is actually telling the truth. Consider the null hypothesis, "the individual is telling the truth," to answer the following questions. ------------ a. What is the probability of a Type I error? -- b. What is the probability of a Type II error? ------------ Explanation: Here the null hypothesis suggests that the person is telling the truth and the alternative hypothesis suggests that the person is not telling the truth.a.Type I error, α = 0.005b.Type II error, β = 1 − 0.95 = 0.05

a. 0.005 -- b. 0.05

A random sample of 24 observations is used to estimate the population mean. The sample mean and the sample standard deviation is calculated as 104.6 and 28.8, respectively. Assume that the population is normally distributed. ---------------- a. Construct the 90% confidence interval for the population mean -- b. Construct the 99% confidence interval for the population mean. -- c. Use your answers to discuss the impact of the confidence level on the width of the interval.

a. 90% Confidence interval = (94.53 , 114.67) -- b. 99% Confidence interval = (88.10 , 121.10) -- c. As the confidence level increases, the interval becomes wider.

A random sample of 80 observations results in 50 successes. --------- a. Construct the 95% confidence interval for the population proportion of successes -- b. Construct the 95% confidence interval for the population proportion of failures.

a. 95% conf interval of SUCCESSES = (0.519 , 0.731) -- b. 95% conf interval of FAILURES = (0.269 , 0.481)

A realtor wants to estimate the mean price of houses in Mission Viejo, California. She collects a sample of 36 recent house sales (in $1,000s), a portion of which is shown in the accompanying table. Assume that the population standard deviation is 100 (in $1,000s) Prices (in 1,000s): (first few) 430, 520, 460, 475... ----------- a. Construct 95% and 98% confidence intervals for the mean price of all houses in Mission Viejo, CA

a. 95% confidence level: (483.4 , 548.7) 98% confidence level: (477.3 , 554.8)

Recently, six single-family homes in San Luis Obispo County in California sold at the following prices (in $1,000s): 549, 449, 705, 529, 639, and 609. -------------- a. Construct the 95% confidence interval for the mean sale price in San Luis Obispo County. -- b. What assumption have you made when constructing this confidence interval?

a. Confidence interval (485.34 , 674.66) -- b. Single family homes in San Luis Obispo follow a normal distribution

You would like to determine if more than 50% of the observations in a population are below 10. At α = 0.05, conduct the test on the basis of the following 20 sample observations: ---- 1st row: 8, 12, 5, 9, 14, 11, 9, 3, 7, 8 2nd row: 12, 6, 8, 9, 2, 6, 11, 4, 13, 10 ----- a. Select the null and the alternative hypotheses. -- b. Calculate the sample proportion. -- c. Calculate the value of the test statistic. -- d. Find the p-value -- e. At the 5% significance level, What is the conclusion? -- f. Interpret the results at α = 0.05.

a. H0: p ≤ 0.50; HA: p > 0.50 -- b. 0.65 -- c. 1.34 -- d. 0.05 ≤ p-value < 0.10 -- e. Do not reject H0 since the p-value is greater than significance level. -- f. We cannot conclude that the population proportion is greater than 0.50.

A study by Allstate Insurance Co. finds that 82% of teenagers have used cell phones while driving (The Wall Street Journal, May 5, 2010). In October 2010, Massachusetts enacted a law that forbids cell phone use by drivers under the age of 18. A policy analyst would like to determine whether the law has decreased the proportion of drivers under the age of 18 who use a cell phone. --------- a. Select the null and the alternative hypotheses to test the policy analyst's objective. -- b-1. Suppose a sample of 200 drivers under the age of 18 results in 150 who still use a cell phone while driving. What is the value of the test statistic? -- b-2. Find the p-value. -- c-1. At α = 0.05, do you reject the null hypothesis? -- c-2. What is the conclusion?

a. H0: p ≥ 0.82; HA: p < 0.82 -- b-1. -2.57 -- b-2. p-value < 0.01 -- c-1. Yes, since the p-value is smaller than significance level. -- c-2. The law has been effective since the p-value is less than the significance level.

It is advertised that the average braking distance for a small car traveling at 65 miles per hour equals 120 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 36 small cars at 65 miles per hour and records the braking distance. The sample average braking distance is computed as 114 feet. Assume that the population standard deviation is 22 feet. ------------------- a. State the null and the alternative hypotheses for the test. -- b. Calculate the value of the test statistic and the p-value -- b1. Find the p-value -- c. Use α = 0.01 to determine if the average breaking distance differs from 120 feet

a. H0: μ = 120; HA: μ ≠ 120 -- b. Test statistic = -1.64 -- b1. p-value ≥ 0.10 -- c. Not significantly different

Select the null and the alternative hypotheses for the following tests: ------------------------- a. Test if the mean weight of cereal in a cereal box differs from 18 ounces. -- b. Test if the stock price increases on more than 60% of the trading days -- c. Test if Americans get an average of less than seven hours of sleep.

a. H0: μ = 18; HA: μ ≠ 18 -- b. H0: p ≤ 0.60; HA: p > 0.60 -- c. H0: μ ≥ 7; HA: μ < 7

An article found that Massachusetts residents spent an average of $860.70 on the lottery in 2010, more than three times the U.S. average (www.businessweek.com, March 14, 2012). A researcher at a Boston think tank believes that Massachusetts residents spend less than this amount. He surveys 100 Massachusetts residents and asks them about their annual expenditures on the lottery ------------------- a. Specify the competing hypotheses to test the researcher's claim. -- b-1. Calculate the value of the test statistic. -- b-2. Find the p-value. -- c. At α = 0.10, what is the conclusion?

a. H0: μ ≥ 860.70; HA: μ < 860.70 -- b1. -0.864 -- b2. p-value ≥ 0.10 -- c. Do not reject H0; there is insufficient evidence to state that the average Massachusetts resident spent less than $860.70 on the lottery in 2010

A consumer advocate researches the length of life between two brands of refrigerators, Brand A and Brand B. He collects data (measured in years) on the longevity of 40 refrigerators for Brand A and repeats the sampling for Brand B. These data are measured in years (1st row: 16, 16, 16, 12 2nd: 14, 20, 16, 19....) ---------------- Assume that μ1 is the mean longevity for Brand A and μ2 is the mean longevity for Brand B. a. Specify the competing hypotheses to test whether the average length of life differs between the two brands. -- b-1. Calculate the value of the test statistic. Assume that σA2 = 4.4 and σB2 = 5.2. -- b-2. Find the p-value -- c. At the 5% significance level, what is the conclusion?

a. H0: μ1 − μ2 = 0; HA: μ1 − μ2 ≠ 0 -- b1. -1.48 -- b2. p-value ≥ 0.10 -- c. Do not reject H0; there is no evidence that the average life differs between the brands.

A sample of 80 results in 30 successes. ---------- a. Calculate the point estimate for the population proportion of successes. -- b. Construct 90% and 99% confidence intervals for the population proportion -- c. Can we conclude at 90% confidence that the population proportion differs from 0.5? -- d. Can we conclude at 99% confidence that the population proportion differs from 0.5?

a. Point estimate = 0.375 -- b. 90% conf. interval = (0.286 , 0.464) 99% conf. interval = (0.236 , 0.514) -- c. Yes, since the confidence interval does not contain the value 0.5 -- d. No, since the confidence interval contains the value 0.5

Let the following sample of 8 observations be drawn from a normal population with unknown mean and standard deviation: 22, 18, 14, 25, 17, 28, 15, 21 ------------- a. Calculate the sample mean and the sample standard deviation -- b. Construct the 80% confidence interval for the population mean. -- c. Construct the 90% confidence interval for the population mean. -- d. What happens to the margin of error as the confidence level increases from 80% to 90%?

a. Sample mean = 20.00 sample standard dev = 4.90 -- b. (17.55 , 22.46) -- c. (16.72 , 23.28) -- d. As the confidence level increases, the margin of error becomes larger.

A study found that consumers are making average monthly debt payments of $983 (Experian.com, November 11, 2010). The accompanying table shows a portion of average monthly debt payments for 26 metropolitan areas -- Washington, D.C.$1,285 Seattle 1,135 Baltimore 1,133, . ... ------------------- a. Calculate the mean and standard deviation for debt payments -- b. Construct the 90% and the 95% confidence intervals for the population mean for debt payments

a. Sample mean = 983.46 Sample mean stand dev = 124. 61 -- b. 90% confidence interval = (941.72 , 1,025.20) 95% confidence interval = (933.12 , 1,033.80)

In order to conduct a hypothesis test for the population proportion, you sample 320 observations that result in 128 successes ------ H0: p ≥ 0.45; HA: p < 0.45 ------ a-1. Calculate the value of the test statistic. -- a-2. Find the p-value -- a-3. At the 5% significance level, What is the conclusion? -- a-4. Interpret the results at a = 0.05 --------------- H0: p = 0.45; HA: p ≠ 0.45 -- b-1. Calculate the value of the test statistic -- b-2. Find the p-value -- b-3. At the 5% significance level, What is the conclusion? -- b-4. Interpret the results at αα = 0.05.

a1. -1.80 -- a2. 0.025 ≤ p-value < 0.05 -- a3. Reject H0 since the p-value is smaller than significance level. -- a4. We conclude that the population proportion is less than 0.45. -- b1. -1.80 -- b2. 0.05 ≤ p-value < 0.10 -- b3. Do not reject H0 since the p-value is greater than significance level -- b4. We cannot conclude that the population proportion differs from 0.45.

What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E = 1.5? The population standard deviation is known to be 10.75.

n = 198

A Type I error occurs when we _______.

reject the null hypothesis when it is actually true

A local courier service advertises that its average delivery time is less than 6 hours for local deliveries. When testing the two hypotheses, Ho: μ ≥ 6 and HA: μ < 6, μ stand for ________

the mean delivery time

When testing the difference between two population means under independent sampling, we use the z distribution if

the population variances are known