MATH& 146 TEST 1

If A and B are independent,

P(A AND B) = P(A)P(B), P(A|B) = P(A) and P(B|A) = P(B).

A health club is interested in knowing how many times a typical member uses the club in a week. They decide to ask every tenth customer on a specified day to complete a short survey including information about how many times they have visited the club in the past week. "Number of visits per week" is what kind of data?

Quantitative - discrete

What does the central limit theorem state with regard to the distribution of sample means?

The central limit theorem states that if samples of sufficient size drawn from a population, the distribution of sample means will be normal, even if the distribution of the population is not normal.

What does the law of large numbers say about the relationship between the sample mean and the population mean?

The law of large numbers says that as sample size increases, the sample mean tends to get nearer and nearer to the population mean.

What are the two essential characteristics of a discrete probability distribution?

The probabilities must sum to 1.0, and the probabilities of each event must be between 0 and 1, inclusive.

Translate this statement about the distribution of a random variable X into words: X ~ (100, 15).

The random variable X has a normal distribution with a mean of 100 and a standard deviation of 15.

The U.S. federal government conducts a survey of high school seniors concerning their plans for future education and employment. One question asks whether they are planning to attend a four-year college or university in the following year. Fifty percent answer yes to this question; that fifty percent is a:

statistic

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: TREATMENT

tableware that is 20 percent smaller than normal

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: SAMPLE

the 1,000 visits drawn for the study

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: PARAMETER

the average expenditure on produce per visit by all the store's customers

In a right-skewed distribution, which is greater?

the mean

The mean of X

μ = np

In a biology class, the scores on the final exam were normally distributed, with a mean of 85, and a standard deviation of five. Susan got a final exam score of 95. Express her exam result as a z-score, and interpret its meaning.

𝑧=(95−85)/5=2.0 Susan's z-score was 2.0, meaning she scored two standard deviations above the class mean for the final exam.

Explain how the rules applying the central limit theorem to sample means, and to sums of a random variable, are similar.

Both rules state that the distribution of a quantity (the mean or the sum) calculated on samples drawn from a population will tend to have a normal distribution, as the sample size increases, regardless of the distribution of population from which the samples are drawn.

How does the central limit theorem apply to sums of random variables?

For a random variable X, the random variable ΣX will tend to become normally distributed as the size n of the samples used to compute the sum increases.

A manager wants to draw a sample, without replacement, of 30 employees from a workforce of 150. Describe how the chance of being selected will change over the course of drawing the sample.

For the first person picked, the chance of any individual being selected is one in 150. For the second person, it is one in 149, for the third it is one in 148, and so on. For the 30th person selected, the chance of selection is one in 121.

You conduct a survey of students to see how many books they purchased the previous semester, the total amount they paid for those books, the number they sold after the semester was over, and the amount of money they received for the books they sold. Which variables in this survey are discrete, and which are continuous?

The discrete variables are the number of books purchased, and the number of books sold after the end of the semester. The continuous variables are the amount of money spent for the books, and the amount of money received when they were sold.

You conduct a survey among a random sample of students at a particular university. The data collected includes their major, the number of classes they took the previous semester, and amount of money they spent on books purchased for classes in the previous semester. If X = student's major, then what is the domain of X ?

The domain of X = {English, Mathematics,....], i.e., a list of all the majors offered at the university, plus "undeclared."

You conduct a survey among a random sample of students at a particular university. The data collected includes their major, the number of classes they took the previous semester, and amount of money they spent on books purchased for classes in the previous semester. If Y = the number of classes taken in the previous semester, what is the domain of Y?

The domain of Y = {0, 1, 2, ...}, i.e., the integers from 0 to the upper limit of classes allowed by the university.

You conduct a survey among a random sample of students at a particular university. The data collected includes their major, the number of classes they took the previous semester, and amount of money they spent on books purchased for classes in the previous semester. If Z = the amount of money spent on books in the previous semester, what is the domain of Z?

The domain of Z = any amount of money from 0 upwards.

You get data from the U.S. Census Bureau on the median household income for your city, and decide to display it graphically. Which is the better choice for this data, a bar graph or a histogram? Explain.

The histogram is a better choice, because income is a continuous variable.

The distribution of results from flipping a fair coin is uniform: heads and tails are equally likely on any flip, and over a large number of trials, you expect about the same number of heads and tails. Yet if you conduct a study by flipping 30 coins and recording the number of heads, and repeat this 100 times, the distribution of the mean number of heads will be approximately normal. How is this possible?

The sample size of 30 is sufficiently large in this example to apply the central limit theorem. This theorem ] states that for samples of sufficient size drawn from a population, the sampling distribution of the sample mean will approach normality, regardless of the distribution of the population from which the samples were drawn.

What are the required characteristics of a binomial experiment?

There are a fixed number of trials. There are only two possible outcomes, and they add up to 1. The trials are independent and conducted under identical conditions.

Two researchers studying vaccination rates independently draw samples of 50 children, ages 3-18 months, from a large urban area, and determine if they are up to date on their vaccinations. One researcher finds that 84 percent of the children in her sample are up to date, and the other finds that 86 percent in his sample are up to date. Assuming both followed proper sampling procedures and did their calculations correctly, what is a likely explanation for this discrepancy?

These results (84 percent in one sample, 86 percent in the other) are probably due to sampling variability. Each researcher drew a different sample of children, and you would not expect them to get exactly the same result, although you would expect the results to be similar, as they are in this case.

Describe how you would draw the continuous probability distribution described by the function 𝑓(𝑥)=110 for 0≤𝑥≤10. What type of a distribution is this?

This is a uniform probability distribution. You would draw it as a rectangle with the vertical sides at 0 and 20, and the horizontal sides at 110 and 0.

Explain why it would not be possible to use random assignment to study the health effects of smoking.

To use random assignment, you would have to be able to assign people to either smoke or not smoke. Because smoking has many harmful effects, this would not be an ethical experiment. Instead, we study people who have chosen to smoke, and compare them to others who have chosen not to smoke, and try to control for the other ways those two groups may differ (lurking variables).

Describe a situation in which you would calculate a parameter, rather than a statistic.

When you use the data from entire population; a manger might calculate the average amount of hours their employees work

If the variable X has the standard normal distribution, express this symbolically.

X ~ N(0,1)

Why is non-response a problem in surveys?

You want the sample of people who take part in a survey to be representative of the population from which they are drawn. People who refuse to take part in a survey often have different views than those who do participate, and so even a random sample may produce biased results if a large percentage of those selected refuse to participate in a survey.

Applying the law of large numbers, which sample mean would expect to be closer to the population mean, a sample of size ten or a sample of size 100?

You would expect the mean from a sample of size 100 to be nearer to the population mean, because the law of large numbers says that as sample size increases, the sample mean tends to approach the population mea.

The mean of a normally-distributed population is 50, and the standard deviation is four. If you draw 100 samples of size 40 from this population, describe what you would expect to see in terms of the sampling distribution of the sample mean.

You would not expect each sample to have a mean of 50, because of sampling variability. However, you would expect the sampling distribution of the sample means to cluster around 50, with an approximately normal distribution, so that values close to 50 are more common than values further removed from 50.

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: POPULATION

all college students

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: POPULATION

all the shopping visits by all the store's customers

Describe how you might draw a random sample of 30 students from a lecture class of 200 students.

assign each student a number from 1 to 200. Then use a random number generator or table of random number to generate 30 numbers between 1 and 200, and select the students matching the random numbers.

The manager of a department store decides to measure employee satisfaction by selecting four departments at random, and conducting interviews with all the employees in those four departments. What type of survey design is this?

cluster

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: EXPERIMENTAL UNITS

each individual college student who participated

formula for finding the kth percentile

i = k/100 (n+1) k= the kth percentile i = the index (ranking or position of the data value) n = the total number of data

X ~ B(n, p)

means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p

X ~ G(p)

means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p

Describe how you might draw a stratified sample of students from a college, where the strata are the students' class standing (freshman, sophomore, junior, or senior).

obtain a roster of students enrolled in the college, including the class standing for each student. Then you would draw a proportionate random sample from within each class

Imagine that the U.S. federal government had the means to survey all high school seniors in the U.S. concerning their plans for future education and employment, and found that 50 percent were planning to attend a 4-year college or university in the following year. This 50 percent is an example of a:

parameter

In a binomial experiment, if p = 0.65, what does q equal?

q = 1 - 0.65 = 0.35

What kind of data is "amount of money spent on produce per visit"?

quantitative - discrete

The study finds that the mean amount spent on produce per visit by the customers in the sample is $12.84. This is an example of a:

statistic

A health club is interested in knowing how many times a typical member uses the club in a week. They decide to ask every tenth customer on a specified day to complete a short survey including information about how many times they have visited the club in the past week. WHAT KIND OF SAMPLING DESIGN IS THIS?

systematic

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: SAMPLE

the 100 college students in the study

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: RESPONSE VARIABLE

the amount of food eaten

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: STATISTIC

the average expenditure on produce per visit by the sample of 1,000

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: DATA

the dollar amounts spent on produce

A grocery store is interested in how much money, on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer's average spending on produce. IDENTIFY: VARIABLE

the expenditure on produce for each visit

In a left-skewed distribution, which is greater?

the mode

A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group IDENTIFY: EXPLANATORY VARIABLE

the size of the tableware

You are reading a research article that refers to "the standard error of the mean." What does this mean?

the standard error of mean is another name for the standard deviation of the sampling distribution of the sample mean. given samples of size n drawn from a population with a standard deviation.

In a symmetrical distribution what will be the relationship among the mean, median, and mode? Discuss.

they will be fairly close to each other

number of standard deviations

value-mean/ standard deviation

formula for finding a percentile of a value in a data set

x+0.5y / n (100)

z-score formula for a population

z = (x - μ)/σ

z-score formula for sample

z= x- mean / s

The standard deviation of X

σ = √npq

In a class of 35 students, seven students received scores in the 70-79 range. What is the relative frequency of scores in this range?

𝑅𝐹=7/35=0.2

If A and B are mutually exclusive,

P(A OR B) = P(A) + P(B) and P(A AND B) = 0.

relative frequency formula

RF = f/n

A professor offers extra credit to students who take part in her research studies. What is an ethical problem with this method of recruiting subjects?

Research subjects should not be coerced into participation, and offering extra credit in exchange for participation could be construed as coercion. In addition, this method will result in a volunteer sample, which cannot be assumed to be representative of the population as a whole.

A professor conducts a telephone survey of a city's population by drawing a sample of numbers from the phone book and having her student assistants call each of the selected numbers once to administer the survey. What are some sources of bias with this survey?

Sources of bias include the fact that not everyone has a telephone, that cell phone numbers are often not listed in published directories, and that an individual might not be at home at the time of the phone call; all these factors make it likely that the respondents to the survey will not be representative of the population as a whole.

For a continuous random variable, why are P(x < c) and P(x ≤ c) equivalent statements?

Because P(x = c) = 0 for any continuous random variable.

You read a newspaper article reporting that eating almonds leads to increased life satisfaction. The study was conducted by the Almond Growers Association, and was based on a randomized survey asking people about their consumption of various foods, including almonds, and also about their satisfaction with different aspects of their life. Does anything about this poll lead you to question its conclusion? Explain.

At least two aspects of this poll are troublesome. The first is that it was conducted by a group who would benefit by the result—almond sales are likely to increase if people believe that eating almonds will make them happier. The second is that this poll found that almond consumption and life satisfaction are correlated, but does not establish that eating almonds causes satisfaction. It is equally possible, for instance, that people with higher incomes are more likely to eat almonds, and are also more satisfied with their lives.

With continuous random variables, we never calculate the probability that X has a particular value, but always speak in terms of the probability that X has a value within a particular range. Why is this?

Because for a continuous random variable, P(x = c) = 0, where c is any single value. Instead, we calculate P(c < x < d), i.e., the probability that the value of x is between the values c and d.

Compute the mean of the following numbers, and report your answer using one more decimal place than is present in the original data: 14, 5, 18, 23, 6

13.2

The midterm grades on a chemistry exam, graded on a scale of 0 to 100, were: 62, 64, 65, 65, 68, 70, 72, 72, 74, 75, 75, 75, 76,78, 78, 81, 83, 83, 84, 85, 87, 88, 92, 95, 98, 98, 100, 100, 740

740 is an outlier because the exam was graded on a scale of 0 to 100. This could be a data entry error so I would have the professor check the exam again

You collect data on the color of cars driven by students in your statistics class, and want to display this information graphically. Which is the better choice for this data, a bar graph or a histogram? Explain.

A bar graph is the better choice, because this data is categorical rather than continuous.

The addition rule:

P(A OR B) = P(A) + P(B) - P(A AND B)

Joe conducts an experiment to see how many times he has to flip a coin before he gets four heads in a row. Does this qualify as a binomial experiment?

No, because there are not a fixed number of trials

In a high school graduating class of 300, 200 students are going to college, 40 are planning to work full-time, and 80 are taking a gap year. Are these events mutually exclusive? Explain.

No, they cannot be mutually exclusive, because they add up to more than 300. Therefore, some students must fit into two or more categories (e.g., both going to college and working full time).

A high school increased the length of the school day from 6.5 to 7.5 hours. Students who wished to attend this high school were required to sign contracts pledging to put forth their best effort on their school work and to obey the school rules; if they did not wish to do so, they could attend another high school in the district. At the end of one year, student performance on statewide tests had increased by ten percentage points over the previous year. Does this improvement prove that a longer school day improves student achievement?

No. The improvement could also be due to self-selection: only motivated students were willing to sign the contract, and they would have done well even in a school with 6.5 hour days. Because both changes were implemented at the same time, it is not possible to separate out their influence.

A popular American television sports program conducts a poll of viewers to see which team they believe will win the NFL (National Football League) championship this year. Viewers vote by calling a number displayed on the television screen and telling the operator which team they think will win. Do you think that those who participate in this poll are representative of all football fans in America? Explain.

No. There are at least two chances for bias. First, the viewers of this particular program may not be representative of American football fans as a whole. Second, the sample will be self-selected, because people have to make a phone call in order to take part, and those people are probably not representative of the American football fan population as a whole.

The multiplication rule:

P(A AND B) = P(A|B)P(B)