Final prep

Use the following for questions 34 to 36: In the past, the percentage of all drivers who use a seat belt was 80%. The chief of The National Highway Traffic Safety Administration believes the population proportion is higher this year. A survey this year found that out of 72 randomly sampled drivers, 61 used a seat belt What is the appropriate alternative hypotheses to be tested? a. 𝐻𝐻:𝑝𝑝> 0.80 b. 𝐻𝐻:𝑝𝑝> 0.85 c. 𝐻𝐻:𝑝𝑝> 61 d. 𝐻𝐻:𝑝𝑝 > 72

(option a) H1: p > 0.80 Explanation: The null hypothesis is that the population proportion of drivers who use a seat belt is 0.80. The alternative hypothesis is that the population proportion is greater than 0.80, as the chief of The National Highway Traffic Safety Administration believes that the proportion is higher this year. Therefore, option a is the correct answer.

If the coefficient for determination for displacement is 66% and the coefficient for determination for horsepower is 48.4%, which is the better predictor of city gas mileage?

Displacement. More variability in the city gas mileage (66%) is explained by displacement than horsepower, which explains 48.4% of the variability in city gas mileage

Customers using a self-service soda dispenser take an average of 12 ounces of soda with an SD of 4 ounces. Assume that the amount would be normally distributed. What is the chance that the next 20 customers will take an average between 13 and 14 ounces? Draw a well-labeled picture of this value.

H0: μ= 12 HA: 13 ≤ μ ≤ 14 use z= (ȳ- μ)/(σ/√n) z1= (13 - 12) / (4 / √20)= 1.118 z2= (14 - 12) / (4 / √20)= 2.236 p(z2 -z1)= 1.118-2.236= 0.119 on your graph, the area between 13 and 14 will be shaded and will be the equivalent of 0.119, the graph will be centered at 12

A large university is planning to build a new parking garage on campus that will be paid for by a larger student parking fee. The student newspaper wanted to determine if the majority of students favor the new garage. They email a random sample of 100 students and find that 57 of them are in favor of the new garage. Conduct an appropriate test of hypothesis. Be sure to include all necessary steps and all relevant information within each step.

Let p be the proportion of all students who favor the new parking garage. Null Hypothesis: H0: p = 0.5 (or less than 50%) Alternative Hypothesis: Ha: p > 0.5 (more than 50%) Step 2: Determine the level of significance. Let alpha (α) = 0.05 be the level of significance. Step 3: Identify the test statistic and the distribution to use. Since we have a sample size of n = 100, which is large enough, and we are testing the proportion of students who favor the new parking garage, we can use the normal distribution to approximate the sampling distribution of the sample proportion. The test statistic is calculated as: z = (phat - p) / sqrt[pq/n], where phat is the sample proportion, p is the hypothesized proportion under the null hypothesis, q = 1 - p, and n is the sample size. Step 4: Calculate the test statistic. phat = 57/100 = 0.57 p = 0.5 q = 1 - p = 0.5 n = 100 z = (0.57 - 0.5) / sqrt[(0.5)(0.5)/100] = 1.6 pvalue: 0.0548. Since the p-value of 0.0548 is greater than the level of significance of 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the majority of students favor the new parking garage.

In statistics we are able to make inference because statistics have a predictable distribution called a sampling distribution. One method of inference we have discussed is the idea of hypothesis testing. Explain how the sampling distribution is used in the process of hypothesis testing

The sampling distribution allows us to quantify the variability in sample statistics. This allows us to calculate a p-value, which is the probability of observing a test statistic that is as extreme as or more extreme than our test statistic assuming the null hypothesis is true. The sampling distribution provides a way to estimate the probability of observing a particular test statistic, which allows us to make decisions about the null hypothesis. The p-value is a way of summarizing this probability and can be used to make a decision about whether to reject or fail to reject the null hypothesis.

The random variable Y follows a binomial distribution with p = 0.3 and n =9. Find the probability that Y = 3. a. 0.267 b. 0.172 c. 0.730 d. 1

a. 0.267 binomial probability: (9!/(3!*6!)*0.3^3*(1-0.3)^6= 0.267

Use the following for questions 29 and 30: A company that makes candy-coated chocolate pieces annually produces a special holiday mix of candy corn colors. They claim that in this mix, 25% of the candies are yellow, 40% are orange, and 35% are white. Suppose we take a random sample of 50 candies from this mix and find the following counts: Color: Yellow Orange White Observed count: 10 23 17 What type of Chi-square test would we use to determine if the company's stated model for the colors is correct? a. Goodness of fit b. Homogeneity c. Independence d. Either b. or c. but not a.

a. Goodness of fit

Studies show that as women grow older they tend to have lower bone density. Which of the following graphs illustrates this point? Graph A: Tightly packed dots around an imaginary line sloping down to the right Graph B: Tightly packed dots around an imaginary line sloping up to the right Graph C: dots evenly scattered over the graph with no visible line of regression

a. Graph A

A correlation of r=0.88 indicates that the graph of the data would show a. Points tightly packed around a line that slopes up to the right. b. Points tightly packed around a line that slopes down to the right. c. Points widely scattered around a line that slopes up to the right. d. Points widely scattered around a line that slopes down to the right.

a. Points tightly packed around a line that slopes up to the right.

Identify the histogram that would provide the each of the summary statistics below. (One histogram will not be used.) a. mean = 4.89, median = 4.83, standard deviation = 7.99 ________ b. mean = 5.01, median = 6.87, standard deviation = 5.49_________ c. mean = 4.96, median = 4.93, standard deviation = 0.96 _________ Image description: graph 1: middle peak centered near 4.8, normally distributed with lower value 1.5 and upper value 7.5 graph 2: peak centered between 5 and 10, normally distributed with lower value of -20 and upper value of +20 graph 3: positively skewed, lower value and peak near 0, upper value 18 graph 4: negatively skewed, lower value 15, upper value and median close to 5

a. graph 2 b. graph 4 c. graph 1

A study was conducted on the amount of time drivers wait for a stoplight to change at a particular intersection. The amount of time spent by 300 drivers was recorded and the resulting data were used to create this boxplot. Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12 The mean amount of time spent at this traffic light was a. greater than the median. b. less than the median. c. about the same as the median. d. It is impossible to tell without the standard deviation

a. greater than the median. positive skew right= mean > median Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12

Use the following for questions 34 to 36: In the past, the percentage of all drivers who use a seat belt was 80%. The chief of The National Highway Traffic Safety Administration believes the population proportion is higher this year. A survey this year found that out of 72 randomly sampled drivers, 61 used a seat belt Calculate the value of the test statistic. a. 0.05 b. 1.00 c. 16 d. 17

b. 1.00 use proportions p= 61/72= .847 P= 0.8 n=72 z*= (0.847 - 0.8) / sqrt(0.8 * 0.2 / 72) z*= .047/.046 z*=1.02

A study was conducted on the amount of time drivers wait for a stoplight to change at a particular intersection. The amount of time spent by 300 drivers was recorded and the resulting data were used to create this boxplot. Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12 26. The median amount of time spent at this traffic light was a. 1.0. b. 2.3. c. 4.0. d. It is impossible to tell without the standard deviation

b. 2.3 Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12

A certain manufacturer claims that 50% of the candies they produce are brown and that candy pieces are randomly placed into bags. Sam plans to buy a large family size bag of these candies and Kerry plans to buy a small fun size bag. Which bag is more likely to have more than 70% brown candies? a. Sam's, because a larger bag is more likely to have a larger proportion of brown candies. b. Kerry's, because there is more variability in proportions of colors among smaller samples. c. Both have the same chance because the bags they buy are both random samples of candy pieces

b. Kerry's, because there is more variability in proportions of colors among smaller samples.

Use the following for questions 34 to 36: In the past, the percentage of all drivers who use a seat belt was 80%. The chief of The National Highway Traffic Safety Administration believes the population proportion is higher this year. A survey this year found that out of 72 randomly sampled drivers, 61 used a seat belt The p-value is 0.145. Are the results statistically significant at the 1% significance level? a. Yes b. No

b. No

Which of the following is not an assumption of the Binomial distribution. a. The trials are independent of each other. b. The variance is equal to the mean of the distribution. c. The probability of a success is the same for all trials. d. The trials each have two possible outcomes (e.g. success, failure)

b. The variance is equal to the mean of the distribution.

Consider the campaign manager in the previous question. Suppose he had based his hypothesis test on a survey of followers of the candidate's twitter account. The resulting responses would be considered a a. volunteer sample. b. convenience sample. c. systematic sample. d. cluster sample

b. convenience sample.

The most important advantage of experiments over observational studies is that a. experiments are usually easier to carry out. b. experiments can give better evidence of causation. c. the placebo effect cannot happen in experiments. d. an observational study cannot have a response variable

b. experiments can give better evidence of causation.

One month before an election, a poll of 360 randomly selected voters showed that 56% were planning to vote for a certain candidate. The campaign manager for that candidate tests the hypotheses 𝐻0= 0.5 vs. 𝐻A> 0.5 based on the newspaper's data. He calculates a test statistic of 2.28 and a p-value of 0.011. What is the best interpretation of this p-value? a. 1.1% of the null distribution is equal to or more extreme than 2.28. b. 1.1% of the null distribution is less than or equal to 2.28. c. 1.1% of the null distribution is greater than or equal to 2.28

c. 1.1% of the null distribution is greater than or equal to 2.28

We would like to construct a 91% confidence interval for the population proportion from a large random sample. What is the appropriate confidence coefficient (Z)? a. 0.09 b. 3.72 c. 1.70 d. 2.62

c. 1.70 the question asks for the coefficient, not the alpha

A study was conducted on the amount of time drivers wait for a stoplight to change at a particular intersection. The amount of time spent by 300 drivers was recorded and the resulting data were used to create this boxplot. Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12 The top 25% of drivers waited over a. 1.3. b. 2.3. c. 4.0. d. It is impossible to tell without the standard deviation.

c. 4.0. Image of box plot: Lower range: 0 Q1: 1.5 Q2: 2.3 Q3: 4 Upper range: 8 Outliers: 9, 12

We use the t-distribution to calculate a confidence interval for the population mean µ. If we increase the sample size from 10 to 20 the interval would become smaller because of a. the change in degrees of freedom. b. the change in standard error. c. both a and b. d. none of the above

c. both a and b.

We would like to test the hypothesis that µ=20 versus the alternative that µ≠20. From a sample of 30 subjects we calculate the test statistic to be t= 2.3. The p-value would be a. 29. b. 0.014. c. 0.986. d. 0.028

d. 0.028

A political scientist wants to know how college students feel about the social security system. She obtains a list of the 3114 undergraduates at her college and mails a questionnaire to 250 students selected at random. Only 100 of the questionnaires are returned. In this study, the rate of non-response would be a. 0.25 b. 0.40 c. 0.75 d. 0.60

d. 0.60 100/250= 0.4 1-0.4= 0.6

Use the following for questions 29 and 30: A company that makes candy-coated chocolate pieces annually produces a special holiday mix of candy corn colors. They claim that in this mix, 25% of the candies are yellow, 40% are orange, and 35% are white. Suppose we take a random sample of 50 candies from this mix and find the following counts: Color: Yellow Orange White Observed count: 10 23 17 If the distribution of colors is really as claimed, how many white candies would we expect in the sample? a. 0.35 b. 3 c. 17 d. 17.5

d. 17.5 .35*50= 17.5

We conduct a regression and find that the least squares line is y=3+5x. This indicates that as the value of x increases by 4 the predicted value of y would increase by: a. 5. b. 8. c. 23. d. 20.

d. 20. slope beta 1=(5*4)

Use the following for questions 39 and 40: Does the type of movie children are watching make a difference in the amount of snacks they will eat? A group of 50 children were randomly assigned to watch either a cartoon or a live action musical (25 to each). Crackers were available in a bowl, and the investigators compared the number of crackers eaten by children while watching the different kinds of movies This study is best described as: a. A placebo controlled experiment. b. A matched pairs experiment. c. A randomized observational study. d. A completely randomized comparative design

d. A completely randomized comparative design

32. The shape of the distribution shown in the histogram at the right would best be described as a. Symmetric b. Bell-shaped c. Bi-modal d. Both a and c, but not b.

d. Both a and c, but not b.

A study is conducted to examine the impact of a new medicine on the cholesterol level in adult males. 20 subjects have their blood tested to examine their cholesterol level. They are then given the medicine for one week and then their cholesterol is again measured. The best way to determine if there was a significant change in blood cholesterol we should a. Create a histogram of the 20 cholesterol levels. b. Create side by side box plots of the cholesterol level before and after the week. c. Calculate the regression to predict the cholesterol after using the cholesterol before. d. Conduct a paired difference t-test for the change in cholesterol.

d. Conduct a paired difference t-test for the change in cholesterol.

We conduct a test of hypothesis using a significance level of 0.05. This implies a. The test has a 5% chance of the alternative hypothesis being true. b. The test has a 5% chance of null hypothesis being true. c. The test has a 95% chance of a type II error. d. None of the above

d. None of the above The significance-level is the probability of eliminating the null-hypothesis of the test when it is really true, which is indicated by α, that is, α is the possibility of making Type I error. Thus, there is a 5% chance of making type I error which is not given in the options.

An engineering team has developed a new engine design that is intended to improve the fuel economy of an automobile. For a family sedan the new engine increased the fuel economy by 18.7 miles per gallon. The researchers conducted a hypothesis test based on a sample of n=40. The resulting p-value was 0.0007. From this we would conclude: a. The result was statistically significant but not likely to be practically significant. b. The result was likely to be practically significant but not statistically significant. c. The result was neither statistically nor practically significant. d. The result was likely to be both statistically and practically significant

d. The result was likely to be both statistically and practically significant

Use the following for questions 39 and 40: Does the type of movie children are watching make a difference in the amount of snacks they will eat? A group of 50 children were randomly assigned to watch either a cartoon or a live action musical (25 to each). Crackers were available in a bowl, and the investigators compared the number of crackers eaten by children while watching the different kinds of movies In this study the explanatory variable is: a. The amount of crackers eaten. b. The children. c. Does the type of movie make a difference in the amount of snacks eaten? d. The type of movie watched

d. The type of movie watched

A student takes a standardized exam. The grader reports the student's standardized score (z-score) as -1.8. This indicates: a. the student scored lower than the average b. the student scored less than one standard deviation from the average c. a mistake has been made in calculating the score, since a standard score can never be negative d. both a and b, but not c

d. both a and b, but not c

The sampling distribution of a statistic is a. the probability that we obtain the statistic in repeated random samples of the same size from the same population. b. the mechanism that determines whether randomization was effective. c. the extent to which the sample results differ systematically from the truth. d. the distribution of values for a statistic in all possible random samples of the same size from the same population

d. the distribution of values for a statistic in all possible random samples of the same size from the same population

An instructor gave an exam for which the highest score earned was only 99 points (instead of 100). He decided to add one point to everyone's score. The effect of this would be that a. the standard deviation would increase by 1. b. the median would change but the mean would not. c. the mean would change but the median would not. d. the standard deviation would not change but the mean and median would increase.

d. the standard deviation would not change but the mean and median would increase.

A correlation is always between 0 and 1

false

If a list of numbers has a mean of 0, then the standard deviation will also be zero.

false

The p-value is the probability that the null hypothesis is correct

false

an outlier will typically fall between q1 and q3

false

selection bias results when the subjects selected for the sample give responses that are different from the truth

false

the poisson distribution is always symmetric

false

A test statistic is used to measure the difference between the observed sample data and what is expected when the null hypothesis is true.

true

a chi square test of homogeneity is used to determine if the distribution of a categorical variable is the same for two or more populations

true

alpha represents the proportion of time a statistical hypothesis test would make a type 1 error, if the test were repeated many times

true

the expected value of the chi-square distribution is equal to the degrees of freedom for the distribution

true