unit 5 stats

Confidence interval

A confidence interval for a model parameter is an interval of values usually of the form estimate (p hat) +-margin of error- Estimates the population proportion

The distribution of wait times for customers at a certain department of motor vehicles in a large city is skewed to the right with mean 23 minutes and standard deviation 11 minutes. A random sample of 50 customer wait times will be selected. Let x¯W represent the sample mean wait time, in minutes. Which of the following is the best interpretation of P(x¯W>25)≈0.10 ?

For a random sample of 50 customer wait times, the probability that the sample mean customer wait time will be greater than 25 minutes is approximately 0.10.

Researchers in plant growth are investigating the proportions of seedlings that sprout under two environmental settings in a lab experiment. Let pˆC represent the sample proportion of seedlings that sprout in garden C, and let pˆH represent the sample proportion of seedlings that sprout in garden H. For random samples of 100 garden C seedlings and 100 garden H seedlings, the sampling distribution of the difference between sample proportions pˆC−pˆH has σpˆC−pˆH≈0.051. Which of the following is the best interpretation of σPˆC−PˆH≈0.051 ?

For all random samples of 100 garden C seedlings and 100 garden H seedlings, the average distance of all possible differences between sample proportions pˆC−pˆHp^C−p^H from the mean of the sampling distribution is approximately 0.051.

Type I, Type II Errors

I. The null hypothesis is true, but we mistakenly reject it.II. The null hypothesis is false, but we fail to reject it.These two types of errors are known as Type I and Type II errors.

Consider two populations of coins, one of pennies and one of quarters. A random sample of 25 pennies was selected, and the mean age of the sample was 32 years. A random sample of 35 quarters was taken, and the mean age of the sample was 19 years.For the sampling distribution of the difference in sample means, have the conditions for normality been met?

No, the conditions for normality have not been met because the sample size for the pennies is not large enough and no information is given about the distributions of the populations.

A river runs through a certain city and divides the city into two parts, west and east. The population proportion of residents in the west who are in favor of building a new bridge across the river is known to be pW=0.30. The population proportion of residents in the east who are in favor of building a new bridge across the river is known to be pE=0.20. Two random samples of city residents of size 50, one sample from the west and one sample from the east, were taken to investigate opinions on the bridge, where pˆW=0.38 and pˆE=0.25 represent the sample proportions. For samples of size 50 from each population, which of the following is the best interpretation of the mean of the sampling distribution of the difference in the sample proportions (west minus east) of residents from the west and east who are in favor of building the bridge?

The mean of the difference of all sample proportions from all random samples of 50 residents from each side of the river is equal to 0.10.

In a certain region of the country, the proportion of the population with blue eyes is currently 17 percent. A random sample of 100 people will be selected from the population. What is the mean of the sampling distribution of the sample proportion of people with blue eyes for samples of size 100 ?

0.17

Conditions

1.) Randomization Condition- If your data come from an experiment, subjectsshould have been randomly assigned to treatments. If you have a surveyyour sample should be a simple random sample of the population.2.) 10% Condition- The sample size, n, must be no larger than 10% of the population3.) Success/Failure Condition- NQ, NP > 10

A sample of manufactured items will be selected from a large population in which 8 percent of the items are defective. Of the following, which is the least value for a sample size that will allow for the sampling distribution of the sample proportion to be assumed approximately normal?

125

A certain company produces fidget spinners with ball bearings made of either plastic or metal. Under standard testing conditions, fidget spinners from this company with plastic bearings spin for an average of 2.7 minutes, while those from this company with metal bearings spin for an average of 4.2 minutes. A random sample of three fidget spinners with plastic bearings is selected from company stock, and each is spun one time under the same standard conditions; let x¯1 represent the average spinning time for these three spinners. A random sample of seven fidget spinners with metal bearings is selected from company stock, and each is likewise spun one time under standard conditions; let x¯2 represent the average spinning time for these seven spinners. What is the mean μ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2 ?

2.7−4.2=−1.52.7−4.2=−1.5

For a certain population of men, 8 percent carry a certain genetic trait. For a certain population of women, 0.5 percent carry the same genetic trait. Let pˆ1 represent the sample proportion of randomly selected men from the population who carry the trait, and let pˆ2p^2 represent the sample proportion of women from the population who carry the trait. For which of the following sample sizes will the sampling distribution of pˆ1−pˆ2 be approximately normal?

200 men and 2,000 women

A tennis shop uses a machine to string tennis racquets to specific tensions requested by their customers. There's variability in the results, so when the machine is set to string at a tension of 606060 pounds, the resulting racquets have a mean tension of 606060 pounds with a standard deviation of 0.50.50, point, 5 pounds. Suppose that we took random samples of 363636 racquets and calculated the sample mean tension from each sample. We can assume that the racquets in each sample are independent. What would be the shape of the sampling distribution of the sample mean tension?

Approximately normal

The distribution of the number of siblings for students at a large high school is skewed to the right with mean 1.8 siblings and standard deviation 0.7 sibling. A random sample of 100 students from the high school will be selected, and the mean number of siblings in the sample will be calculated.Which of the following describes the sampling distribution of the sample mean for samples of size 100 ?

Approximately normal with standard deviation less than 0.7 sibling

A group of health researchers survey a SRS of 300 teenagers, of whom 120 have low weekly outdoor activity levels (<5<5is less than, 5 hours a week) and 180180180 have high levels (\geq 5≥5is greater than or equal to, 5 hours a week). Suppose that 35\%35%35, percent of teenagers with low outdoor activity levels have allergies, and that 41\%41%41, percent of teenagers with high outdoor activity levels have allergies. The researchers want to see the difference between the sample proportions (\hat{p}_\text{H}-\hat{p}_\text{L})(p^​H​−p^​L​)left parenthesis, p, with, hat, on top, start subscript, start text, H, end text, end subscript, minus, p, with, hat, on top, start subscript, start text, L, end text, end subscript, right parenthesis of teenagers with allergies.

Approximately normal, because we expect 42 successes and 78 failures from teenagers with low levels, and 73.8, point, and 106.2, point from teenagers with high levels, and all of these counts are at least 10.

The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85.Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?

Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.

Sampling distribution

Different random samples give different values for a statistic. The sampling distribution model shows the behavior of the statistic over all the possible samples for the same size n.

A manufacturer of computer monitors estimates that 4 percent of all the monitors manufactured have a screen defect. Let pd represent the population proportion of all monitors manufactured that have a screen defect. For the sampling distribution of the sample proportion for samples of size 100, μPˆd=0.04. Which of the following is the best interpretation of μPˆd=0.04 ?

For all samples of size 100, the mean of all possible sample proportions of monitors manufactured that have a screen defect is 0.04.

Xavi wants to measure the impact of using water for irrigation on a stream's flow rate. The flow rates both upstream and downstream of the irrigation point are normally distributed. For each location, Xavi selects a random time each month and measures the flow rate in liters per second. After he has 121212 samples per location, he will look at the difference \left( \text{upstream} - \text{downstream} \right)(upstream−downstream)left parenthesis, start text, u, p, s, t, r, e, a, m, end text, minus, start text, d, o, w, n, s, t, r, e, a, m, end text, right parenthesis in their sample means. What do we know about the shape of the sampling distribution of \bar{x}_\text{U} - \bar{x}_\text{D}xˉU​−xˉD​x, with, \bar, on top, start subscript, start text, U, end text, end subscript, minus, x, with, \bar, on top, start subscript, start text, D, end text, end subscript, and why?

It's exactly normal, because both populations are normally distributed.

Sampling distribution model for a proportion

Mean is P and the standard deviation is square root of p*q/n

Recipes for the same type of cookies can vary in terms of ingredients and baking times. From a collection of chocolate chip cookie recipes, a baker randomly selected 5 recipes. From a collection of oatmeal raisin cookie recipes, the baker randomly selected 4 recipes. The mean baking times, in minutes, for each sample were recorded as x¯C and x¯O, respectively.What is the correct unit of measure for the standard deviation of the sampling distribution of x¯C−x¯O?

Minutes

Child psychologists study the time, in months, that it takes for infant boys and girls to say their first words. For a certain population, the distributions of time for both populations have the same mean and have the same standard deviation, μ=8 months and σ=1.4 months. Two independent random samples of infant boys and girls were taken, and the time it took for the infants in each sample to say their first words was recorded. The summary statistics for the number of months are shown in the following table.Riley, a child psychologist, claims that for all samples of size 40 from the population of infant boys and all samples of size 40 of newborn girls, the mean of the sampling distribution of the difference in sample means (boys minus girls) is 0.2 month. Is Riley correct?

No, the mean is 8−8=0 months.

An apple producer sells their product in bags that each contain 161616 apples. These apples have a mean weight of 808080 grams and a standard deviation of 555 grams. Suppose that each bag represents an SRS of apples, and we calculate the sample mean weight \bar xxˉx, with, \bar, on top of the apples in each bag.

mean=80 sd=1.25

Individual bottles of water are filled by a machine at a factory with an amount of water that is approximately normal with a mean of 505\,\text{mL}505mL505, start text, m, L, end text and a standard deviation of 10\,\text{mL}10mL10, start text, m, L, end text. A random sample of 161616 bottles is selected for a quality inspection. What is the probability that the mean amount of water in these 161616 bottles \bar xxˉx, with, \bar, on top is within 5\,\text{mL}5mL5, start text, m, L, end text of the population mean?

P(500<x<510)=0.95

Beatriz takes a simple random sample of 350 orders from the more than 5000 total orders to her store and finds that 12% percent of the sampled orders were returned. Assuming that it is really 9% percent of all orders to her store which were returned, what is the approximate probability that more than 12% percent of the sampled orders would have been returned?

P(p^​>0.12)≈0.02

Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution model of the samplemean (and proportion) from a random sample is approximately Normal for large n, regardless of thedistribution of the population, as long as the observations are independent.

The distribution of the commute times for the employees at a large company has mean 22.4 minutes and standard deviation 6.8 minutes. A random sample of n employees will be selected and their commute times will be recorded.What is true about the sampling distribution of the sample mean as n increases from 2 to 10 ?

The mean does not change, and the variance decreases.

At a national convention attended by many educators, about 30 percent of the attendees are from the northeast. Of all the attendees of the national convention, 25 will be selected at random to receive a free book. What are the mean and standard deviation of the sampling distribution of the proportion of attendees from the northeast for samples of size 25 ?

The mean is 0.3 and the standard deviation is 0.3(0.7)25√0.3(0.7)/25.

The mean age of the employees at a large corporation is 35.2 years, and the standard deviation is 9.5 years. A random sample of 4 employees will be selected.What are the mean and standard deviation of the sampling distribution of the sample mean for samples of size 4 ?

The mean is 35.2, and the standard deviation is 9.5/2.

Sampling distribution model for a mean

The mean is mu and the standard deviation is the population standard deviation (theta)/square root of n

A survey of 100 randomly selected dentists in the state of Ohio results in 78% who would recommend the use of a certain toothpaste. The population proportion is known to be p=0.72. For samples of size 100, which of the following best interprets the mean of the sampling distribution of sample proportion of dentists in the state of Ohio who would recommend the use of a certain toothpaste?

The mean of all sample proportions from all random samples of 100 dentists in the state of Ohio is equal to 0.72.

A sports magazine reports that the mean number of hot dogs sold by hot dog vendors at a certain sporting event is equal to 150. A random sample of 50 hot dog vendors was selected, and the mean number of hot dogs sold by the vendors at the sporting event was 140.For samples of size 50, which of the following is true about the sampling distribution of the sample mean number of hot dogs sold by hot dog vendors at the sporting event?

The mean of the sampling distribution of the sample mean is 150 hot dogs.

For a weekly town council meeting in a certain town, the distribution of the duration of the meeting is approximately normal with mean 53 minutes and standard deviation 2.5 minutes. For a weekly arts council meeting in the same town, the distribution of the duration of the meeting is approximately normal with mean 56 minutes and standard deviation 5.1 minutes. Let x¯1 represent the average duration, in minutes, of 10 randomly selected town council meetings, and let x¯2 represent the average duration, in minutes, of 10 randomly selected arts council meetings.Which of the following is the best reason why the sampling distribution of x¯1−x¯2 can be modeled by a normal distribution?

The population distributions are approximately normal.

In one city, 75 percent of residents report that they regularly recycle. In a second city, 90 percent of residents report that they regularly recycle. Simple random samples of 75 residents are selected from each city. Which of the following statements is correct about the approximate normality of the sampling distribution of the difference in sample proportions of residents who report that they regularly recycle?

The sampling distribution is not approximately normal because although the sample size for the first city is large enough, the sample size for the second city is not large enough.

According to a recent survey, 47.9 percent of housing units in a large city are rentals. A sample of 210 housing units will be randomly selected. Which of the following must be true for the sampling distribution of the sample proportion of housing units in the large city that are rentals to be approximately normal?

The values of 210(0.479) and 210(0.521) must be at least 10.

Sampling variability/Sampling error

The variability we expect to see from one random sample to another. It is sometimes calledSampling error, but sampling variability is the better term.

Standard error

When we estimate the standard deviation of a sampling distribution using statistics found from the data, the estimate is called a standard error.- SE = square root of p1q1/n1

According to a Gallup poll in 2006, about 10\%10%10, percent of Americans said they were "very afraid" of flying on an airplane. Suppose that we took random samples of n=40n=40n, equals, 40 people from this population and computed the proportion of people in each sample who were very afraid of flying. Which of the following distributions is the best approximation of the sampling distribution of the proportion of people who were very afraid of flying? Each distribution uses the same scale.

left to right and 0.1 not really in the center

Suppose 65% of the 310,000 citizens in a county are registered to vote. A not-for-profit organization takes an SRS of 100100100 of these citizens and finds that 67\%67%67, percent of those sampled are registered to vote. The organization plans on taking repeated samples like this. What are the mean and standard deviation of the sampling distribution of the proportion of citizens who are registered to vote?

mean p=0.65 sdp=squarrt0.65(0.35)/100

A geology teacher takes her class on an annual field trip to analyze rock samples. They use randomized sets of coordinates to select samples from two regions. Suppose they examine a sample of 707070 rocks from region A, which is composed of 15\%15%15, percent igneous rock. Then they examine a sample of 606060 rocks from region B, which is composed of 20\%20%20, percent igneous rock. Then they look at the difference between the sample proportions (\hat{p}_\text{A}-\hat{p}_\text{B})(p^​A​−p^​B​)left parenthesis, p, with, hat, on top, start subscript, start text, A, end text, end subscript, minus, p, with, hat, on top, start subscript, start text, B, end text, end subscript, right parenthesis. What are the mean and standard deviation of the sampling distribution of \hat{p}_\text{A}-\hat{p}_\text{B}p^​A​−p^​B​p, with, hat, on top, start subscript, start text, A, end text, end subscript, minus, p, with, hat, on top, start subscript, start text, B, end text, end subscript?

mean=-0.05 sd=square root 0.15(0.85)/70+0.2(0.80/60

Amber and Shinji both play a single-player game where they try to survive as many rounds as possible without crashing a bird into an obstacle. Here's the historical data for how many rounds each player survives in a single attempt at this game: PlayerShapeMeanStd. dev.AmberNormal35.535.535, point, 57.17.17, point, 1ShinjiNormal30.030.030, point, 08.68.6 They decide to each play 101010 attempts and calculate the sample mean of how many rounds they survive. They will then look at the difference in their sample means \left( \bar{x}_\text{A} - \bar{x}_\text{S} \right)(xˉA​−xˉS​)left parenthesis, x, with, \bar, on top, start subscript, start text, A, end text, end subscript, minus, x, with, \bar, on top, start subscript, start text, S, end text, end subscript, right parenthesis. What are the mean and standard deviation (in rounds) of the sampling distribution of \bar{x}_\text{A}-\bar{x}_\text{S}xˉA​−xˉS​x, with, \bar, on top, start subscript, start text, A, end text, end subscript, minus, x, with, \bar, on top, start subscript, start text, S, end text, end subscript?

mean=5.5 sd=squarert (7.1)squrd/10+(8.6)squrd/10

A study reported that finger rings increase the growth of bacteria on health-care workers' hands. Research suggests that 31 percent of health-care workers who wear rings have bacteria on one or both hands, and 27 percent of health-care workers without rings have bacteria on one or both hands. Suppose that independent random samples of 100 health-care workers wearing rings and 100 health-care workers not wearing rings are selected. What is the standard deviation of the sampling distribution of the difference in the sample proportions (wear rings minus does not wear rings) of health-care workers having bacteria on one or both hands?

√0.31(0.69)100+0.27(0.73)100

A fair six-sided die will be rolled fifteen times, and the numbers that land face up will be recorded. Let x¯1 represent the average of the numbers that land face up for the first five rolls, and let x¯2 represent the average of the numbers landing face up for the remaining ten rolls. The mean μ and variance σ2 of a single roll are 3.5 and 2.92, respectively. What is the standard deviation σ(x¯1−x¯2) of the sampling distribution of the difference in sample means x¯1−x¯2?

√2.92/5+2.92/10

For two populations of rabbits, R and S, the proportions of rabbits with white markings on their fur are given as pR and pS, respectively. Suppose that independent random samples of 50 rabbits from R and 100 rabbits from S are selected. Let pˆR be the sample proportion of rabbits with white markings from R, and let pˆS be the sample proportion of rabbits with white markings from S. What is the standard deviation of the sampling distribution of pˆR−pˆS ?

√pR(1−pR)/50+pS(1−pS)/100