incorrect problems on stats medic

the defintion on sampling distribution

A sampling distribution is the distribution of a statistic calculated from all possible samples of the same size from the same population.

For which of the following samples would it be appropriate to use tt-procedures for inference for the population mean?

Sample I shows no strong skew or outliers, and sample III has a sample size at least 30.

After completing a statistical analysis of a survey of 40 students, the principal of North High School made the following conclusion: reject the null hypothesis; there is convincing evidence that more than 50% of students support a schedule change to have lunch occur earlier in the day. Which error could have been committed?

Type I error is rejecting the null hypothesis when the null hypothesis is actually true.

Twenty volunteers with high cholesterol were selected for a trial to determine whether a new diet reduces cholesterol. The volunteers were given a low-carb, low-calorie diet. After 8 weeks of the diet, the average cholesterol of the volunteers dropped a significant amount. This study is an example of

an uncontrolled experiment: All volunteers received the diet, so there is no control group to compare results. A control group would be a group that was instructed to eat their usual diet during the 8 week period, for example.

if zero is contained in the interval that means...

it is plausible that there is no difference

6. A random sample of 95 vehicles is taken from a large parking lot at an office park. Below is the type of each vehicle, and whether it is owned or leased by the driver. A chi-square test will be conducted to determine if there is an association between type of vehicle and method of obtaining the vehicle. Which of the following expressions represents the expected count of leased SUVs?

Expected counts for two-way tables are found using \frac{(row total)(column total)}{table total}tabletotal(rowtotal)(columntotal)​.

A social media developer wants to determine if the proportion of teenagers who use Facebook is the same as the proportion of teenagers who use Snapchat. She takes a random sample of 100 teenagers and finds that 75 of the 100 students use Facebook and 89 of the 100 students use Snapchat. Would it be reasonable for the social media developer to construct a 95% confidence interval for the true difference in proportion of teenagers that use Facebook and Snapchat?

In order to perform inference for a difference in proportions, the two samples must be independent. In this context, there is only a single sample of 100 students.

Redundant monitoring means that a signal is sent in multiple forms or paths to reach its intended target. For example, an alarm may be set up to send multiple signals to a security company, each independent of each other in case of failure along one path. If an alarm sends off four signals down different paths, and each has a 0.65 probability of reaching the security company, what is the probability that at least one signal successfully reaches the security company?

P(At least 1 success) = 1 - P(no successes) = 1 - 0.35^4 = 0.9851−0.354=0.985

18. According to the Guinness Book of World Records, a woman from Russia, Mrs. Vassilyeva, had 69 children between the years 1725 to 1765. She had 16 pairs of twins, 7 sets of triplets, and 4 sets of quadruplets. Suppose one of the births is randomly selected. Given that Mrs. Vassilyeva gave birth to at least 3 children (triplets), what is the probability that she gave birth to quadruplets?

P(Mrs. Vassilyeva gave birth to quadruplets | she gave birth to at least triplets) = 4/11. Of the 11 times she gave birth to at least triplets, 4 of those births were quadruplets.

12. A pollster asked 100 people "If money was not a factor, how many children would you like to have?" The results and their frequencies are shown in the (estimated) probability distribution function table below.

P(X = 0) = 1 - 0.26 - 0.29 - 0.15 - 0.05 = 0.25. \mu _{x}μx​ = (0)(0.25) + (1)(0.26) + (2)(0.29) + (3)(0.15) + (4)(0.05) = 1.49 children. P(X < 1.49) = P(X = 0 or X = 1) = 0.25 + 0.26 = 0.51.

10. The heights of all adult males in Croatia are approximately normally distributed with a mean of 180 cm and a standard deviation of 7 cm. The heights of all adult females in Croatia are approximately normally distributed with a mean of 158 cm and a standard deviation of 9 cm. If independent random samples of 10 adult males and 10 adult females are taken, what is the probability that the difference in sample means (males - females) is greater than 20 cm?

The sampling distribution of \bar x_1-\bar x_2xˉ1​−xˉ2​ has a mean of 22 and standard deviation of 3.6055. P(z > -0.5547) = 0.7104.

Donner Summit, California, is a popular ski resort area. Over the past 60 years, the annual snowfall totals of Donner Summit have followed a distribution that is strongly skewed right with a mean of 404 inches and a standard deviation of 129 inches. If many samples of size 9 were taken, which of the following would best describe the shape of the sampling distribution of \bar xxˉ?

The small sample size means that the shape of the sampling distribution will be similar to the shape of the population distribution, but the central limit theorem states that the sampling distribution will tend towards normal as the sample size increases.

5. A study intends to estimate a population mean with an unknown population standard deviation and a sample size of 15. Which of the following is closest to the appropriate critical value to create a 98% confidence interval?

The t-distribution is used with df=14 and a 98% confidence level (0.01 upper tail probability).

8. Do stain-polyurethane mixes protect wood as well as stain and polyurethane applied in separate coats? Five types of wood will be used, with two boards of each type of wood. One board of each type of wood (randomly selected) will have the stain-polyurethane mix applied to it, and the other will have stain and polyurethane applied in separate coats. Each board will have water poured on it, and the amount of water retained will be measured. Which significance test is most appropriate?

This is a matched pairs experimental design where the statistic is the mean of the differences for each pair.

At a summer camp, 72% of the campers participate in rope climbing and 26% participate in canoeing. 83% of the campers participate in rope climbing, canoeing, or both. What is the probability that a randomly selected camper participates in both rope climbing and canoeing?

Using the "OR" probability formula: 0.83 = 0.72 + 0.26 - P(R and C) P(R and C) = 0.15.

A significance test was conducted using the hypotheses H_0: \mu_N-\mu_H=0, H_a: \mu_N-\mu_H < 0H0​:μN​−μH​=0,Ha​:μN​−μH​<0 where \mu_NμN​ is the true mean number of fouls called during games played at neutral sites and \mu_HμH​ is the true mean number of fouls called during games played at the home team's stadium with a resulting P-value of 0.24. Which of the following is an accurate interpretation of this P-value?

(D) If the null hypothesis is true, there is a 24% probability of getting a sample difference in means as far or farther below 0 as the difference found in the samples.

Forty adult males volunteered to participate in an experiment. Half of them are randomly assigned to take a caffeine supplement before working out, and the other half are assigned to take a placebo before completing the same workout. During the workout, heart rate monitors will be used to measure each participant's heart rate. The study found that those who took a caffeine supplement had significantly higher average pulse rates during the workout. What conclusion can be drawn from the study?

Caffeine supplements will, on average, raise the heart rates of adult males similar to those in this study during a workout similar to the one performed in the experiment. The study used random assignment, so causation can be inferred, but the study used volunteers, not random selection, so the results can only be inferred for those similar to the subjects of the study.

6. A newspaper plans to conduct a survey for the upcoming presidential election in order to estimate the proportion of the population, p, who supports a certain candidate. What is the smallest sample size needed to obtain an estimate that is within 4% of the true proportion p at the 96% confidence level?

since we don't have an estimate for p, we use p=0.5, which gives us the largest possible margin of error, and the critical value for 96% confidence is z* = 2.054. Use algebra to solve for n and then round up to 660.

According to a study, 91% of all adults have a cell phone. An employee of a cell phone company attends a community event and has a special offer to give to first time cell phone owners. If she randomly selects adults in attendance at the event and cell phone ownership is independent from adult to adult, what is the probability that she asks 21 adults total to find the first one that does not own a cell phone?

If the employee asks 21 adults before finding one that does not own a cell phone, this means that the first 20 adults she asked DID have a cell phone, followed by the 21st person who DID NOT have a cell phone. Assuming cell phone ownership is independent from adult to adult gives a probability of (0.91)20(0.09).

When playing the card game Blackjack, multiple decks are used and reshuffled often so that the outcomes of the cards dealt are approximately independent. When a player receives two cards that are a combination of an ace and a face card, this is called a "natural blackjack" and automatically wins. A natural blackjack should occur in 4.5% of the rounds played. What is the probability that a player plays 20 rounds of Blackjack and gets two or more natural blackjacks?

the number of natural blackjacks (X) follows a binomial distribution with n=20, p=0.045. P(X\geq 2) = 1 - P(X\leq 1) = 1 - 0.773 = 0.227P(X≥2)=1−P(X≤1)=1−0.773=0.227.

9. Using data collected from 1981 to 2010 for Ann Arbor, MI (GO BLUE!), the average "high" temperature for days in July has a mean of 28.9° Celsius with a standard deviation of 3.3° Celsius. What are the mean and standard deviation if the temperatures are converted to degrees Fahrenheit?

SD= square root of np(1−p)​ =sqrt{(50)(0.03)(0.97)} = 1.206.

A recent article claimed that women are waiting longer to have their first child. The article estimates that the average age of first-time mothers is 26 years old, which is up from 21 years old in 1970. The margin of error for the estimate was 1.5 years. Based on the estimate and the margin of error, which of the following is an appropriate conclusion?

The margin of error of 1.5 years indicates that we are confident the population mean age is captured in the interval 26 - 1.5 = 24.5 to 26 + 1.5 = 27.5 years old. Therefore, it is plausible that the average age of first-time mothers is 27 years old.

A company intends to collect a random sample of size nn from a population with population proportion of interest pp. Which of the following situations would result in the smallest standard deviation of the sampling distribution of \hat pp^​?

The population proportion p\ne 0.5p≠0.5, and the sample size is 2n2n.

P(Power) + P(Type II Error) = 1, so P(Type II Error) = 1 - P(Power) = 1 - 0.9228 = 0.0772.

The probability of making a Type I error is equal to \alphaα = 0.01, the significance level of the test.

A random sample of 18 adults, chosen from the 1500 adults in the town, took a survey asking their opinion on a recent property tax change. 25% of those who responded said they were in favor of the change. The company running the survey wants to construct a confidence interval estimating the proportion of all adults in the town who support the change. Which of the conditions for inference have been satisfied? I. Random condition II. Normal condition III. 10% condition

The sample was a random sample and the sample (18) is less than 10% of the population (1500). The normal condition is not satisfied as n\hat pnp^​ = 18(0.25) = 4.5 < 10.

Assuming that all conditions for inference are met, which of the following is the appropriate test statistic for testing the null hypothesis that the slope of the population regression line equals 0?

The test statistic is t = (13.4862 - 0) / 0.5310 = 25.40.

According to school records, your school's softball team wins 62% of the time when they play a game on their home field and 26% of the time when they play at the other team's field. This season, they play 45% of their games at their home field. Assuming this team wins at the same pace as previous teams, what is the probability that they win a randomly selected game this season?

This is the sum of the probabilities that the game is at their home field and they win and that the game is at the other team's field and they win. (0.45)(0.62) + (0.55)(0.26) = 0.422

A company advertises two car tire models. The number of thousands of miles that the standard model tires last has a mean \mu_S = 60μS​=60 and standard deviation \sigma_S = 5σS​=5. The number of miles that the extended life tires last has a mean \mu_E = 70μE​=70 and standard deviation \sigma_E = 7σE​=7. If mileages for both tires follow a normal distribution, what is the probability that a randomly selected standard model tire will get more mileage than a randomly selected extended life tire?

Use the random variable E - S, which will follow a normal distribution with a mean of 70 - 60 = 10 and SD = \sqrt{5^2+7^2}52+72​. P(E - S < 0) = 0.123

Major League Baseball (MLB) has recently been evaluating the timing of various events during games in an effort to improve the pace of a game. MLB wants to know how long a mound visit, defined as when a coach pauses the game to visit the pitcher on the mound, takes on average. MLB randomly selects 100 games over the course of a season, and records the length, in seconds, of every mound visit that occurs in that game. This sample of mound visits can be best described as a

cluster sample: The games (clusters) are randomly selected, and all of the mound visits in the selected games are in the sample.

An analysis of 8 used trucks listed for sale in the 48076 zip code finds that the power model ln(\hat{price})=3.748-0.1395ln(miles)ln(price^​)=3.748−0.1395ln(miles), for price (in thousands of dollars) and miles driven (in thousands), is an appropriate model of the relationship. If a used truck has been driven for 47,000 miles, which of the following is closest to the predicted price for the truck?

ln(price^​)=3.748−0.1395ln(47) = 3.21. e^{3.21} = 24.8e3.21=24.8 thousand dollars

The distribution of this approximate sampling distribution will be closer to approximately normal than the distribution of the population due to the Central Limit Theorem, will have the same mean as the distribution of the population ($150), and the standard deviation will be \($50/\sqrt{25}=$10\).

sample of 50 to 100