Econ

Airline travelers should be ready to be more flexible as airlines once again cancel thousands of flights this summer. The Coalition for Airline Passengers Rights, Health, and Safety averages 400 calls a day to help stranded travelers deal with airlines. Suppose the hotline is staffed for 16 hours a day. Calculate the average number of calls in a 60 minute interval.

25

Find zα/2 for 99% confidence level used in estimating the population mean.

2.575

A _____________ random variable is defined as the number of successes achieved in the n trials of a Bernoulli process.

Binomial

Inferential statistics which we studied in this part has two main applications: confidence intervals and hypotheses testing. What is the key difference between these approaches of inferential statistics?

Confidence intervals analysis does not require any prior knowledge that we want to test. Hypothesis testing requires to have some prior knowledge that we test using data.

Which of the following statements is MOST accurate about a Poisson random variable?

It counts the number of successes in a specified time or space interval.

A function that assigns numerical values to the outcomes of a random experiment is called a

Random Variable

In the following scenarios, indicate those that describe a binomial random variable.

The number of mortgage applicants who receive a loan in a sample of 10 applicants.

A survey is designed to collect data on students' evaluations of their instructor's teaching performance. Which of the following situations most likely results in social-desirability biases?

The survey is administered in class with the instructor in the room the entire time.

If the value of the test statistic falls in the rejection region, then the p-value must be less than alpha.

True

In general a reader of a scientific research would prefer to see smaller margins of error than larger margins of errors since results will be more precise.

True

The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution of the sample mean becomes

closer to a normal distribution

For a continuous random variable X, the function used to find the area under f(x) up to any value x is called the

cumulative distribution function (CDF) and z-table is an example of a CDF for a standard normal random variable.

The probability distribution of a continuous random variable is called its probability

density function (or PDF).

Nonresponse bias occurs when

those responding to a survey or poll differ systematically from the non-respondents.

The two main components of a confidence interval are the point estimate (sample mean) and the margin of error.

true

Suppose you were told that the delivery time of your new washing machine is equally likely over the time period 9 am to noon. If we define the random variable X as delivery time, then X follows the

Continuous uniform distribution

The formula for a mean confidence interval is valid only if sample mean (approximately) follows a

normal distribution

In class we discussed two famous cases when polling companies could not predict well elections because of different sample biases. Consider second case we discussed, 2016 election case in the US. In that case what kind of biases were suggested to be present?

three: sample selection, non-response bias, and social desirability bias

Ideally a researcher should be objective and be indifferent in conclusion of the test. However in practice a researcher starts a research with a certain belief that existing knowledge is not correct and is happy when

null hypothesis is rejected

We transform X into Z by subtracting from X its mean and dividing by its standard deviation (it follows from Theorem 1 and 2 of Lecture 6).

true

Consider an example of type 1 and type 2 error which could happen in a court decision. What is type 2 error?

type 2 error is concluding that a person is innocent given that a person is actually guilty.

A random variable X follows the continuous uniform distribution with a lower bound of 2 and an upper bound of 8. What is the height of the density function f(x)?

1/6

The formula σ/{n calculates

Standard error of sample mean

A marketing firm is considering making up to three new hires. Given its specific needs, the management feels that there is a 60% chance of hiring at least two candidates. There is only a 15% chance that it will not make any hires and a 10% chance that it will make all three hires. What is the probability that firm hires only one hire?

.25

Suppose sampling is very costly (think about crash tests for cars) and suppose that the characteristic of interest is normally distributed in the population of all objects of interest. What is the sampling distribution for the sample mean from a random sample of size n=3?

It is normal distribution since population from which we sample is normal (Theorem 3 of Lecture 7)

One way to battle social-desirability bias is to use

online surveys

It is advertised that the average braking distance for a small car traveling at 50 miles per hour equals 90 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 49 small cars at 50 miles per hour and records the braking distance. The sample average braking distance is computed as 98 feet. Assume that the population standard deviation is 20 feet. Calculate the value of the test statistic for null hypothesis H0: μ=90 when tested against alternative hypothesis Ha: μ≠90. Based on test statistic you found in previous problem, find p-value for this test. (Hint: If you follow my approach in class, then first you find p-value over two using z table, and then multiply it by two to find p-value)

0.0052

A random sample is drawn from a normally distributed population with mean μ = 15 and standard deviation σ = 1.5. Suppose the sampling distribution of the sample mean is normally distributed with n = 40, then calculate the probability that the sample mean is more than 15.5.

0.0174

Sikhism, a religion founded in the 15th century in India, is going through turmoil due to a rapid decline in the number of Sikh youths who wear turbans. The tedious task of combing and tying up long hair and a desire to assimilate has led to approximately 25% of Sikh youths giving up the turban. What is the probability that exactly two in a random sample of six Sikh youths wear a turban?

0.0330

Airline travelers should be ready to be more flexible as airlines once again cancel thousands of flights this summer. The Coalition for Airline Passengers Rights, Health, and Safety averages 400 calls a day to help stranded travelers deal with airlines. Suppose the hotline is staffed for 16 hours a day. What is the probability of exactly twenty calls in a 60-minute interval?

0.0519

In a rural town, the average high school teacher annual salary is $43,000. Let salary be normally distributed with a standard deviation of $20,000 (You may find it useful to reference the z-table that you should have with you. Round your answers to 4 decimal places.) What percentage of high school teachers make more than $75,000?

0.0548

Let X be normally distributed with mean μ = 5 and standard deviation σ = 10. (You may find it useful to reference the z-table that you should have with you. Round your answers to 4 decimal places.) Find P(X ≤ 0)

0.3085

At a local community college, 70% of students who enter the college as freshmen go on to graduate. Three freshmen are randomly selected. What is the probability that all of them graduates from the local community college?

0.343

Suppose your boss gives you a task to solve some interesting problem. If you will think hard one hour there is 25% chance you will find a solution. What is the probability that you will find solution during the fourth hour of work?

0.75*0.75*0.75*0.25

You can publish your confidence interval for population mean for any of these confidence levels: 0.90, 0.95, or 0.99. Which of these confidence levels will allow you to make the most precise scientific statement (precise means smaller margin of error or narrower confidence intervals).

0.90

The arrival time of an elevator in a 12-story dormitory is equally likely at any time during the next 2 minutes. Calculate the expected arrival time.

1

Suppose your boss gives you a task to solve some interesting problem. If you will think hard one hour there is 50% chance you will find a solution. Your boss asks to provide him with your expectation for the time required to solve this problem (find E(X)).

2

It is advertised that the average braking distance for a small car traveling at 50 miles per hour equals 90 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 49 small cars at 50 miles per hour and records the braking distance. The sample average braking distance is computed as 98 feet. Assume that the population standard deviation is 20 feet. Calculate the value of the test statistic for null hypothesis H0: μ=90 when tested against alternative hypothesis Ha: μ≠90.

2.8

Sikhism, a religion founded in the 15th century in India, is going through turmoil due to a rapid decline in the number of Sikh youths who wear turbans. The tedious task of combing and tying up long hair and a desire to assimilate has led to approximately 25% of Sikh youths giving up the turban. How to find C62, i.e. number of ways that 2 persons will wear a turban out of 6 sampled people?

6!/(6-2)!2!

At a local community college, 70% of students who enter the college as freshmen go on to graduate. 100 freshmen are randomly selected. What is the expected number that will graduate?

70

In class I discussed with you reason or reasons for why statisticians are especially interested in normal distribution. Which is true about information I provided you in lecture.

I gave two reasons. Histograms of many variables in real world remind normal distribution (heights, weights, investment returns, sizes of planets, etc.) and the fact that sampling errors turn out to have also normal distribution if sample size is large enough (n>30).

It is advertised that the average braking distance for a small car traveling at 50 miles per hour equals 90 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 49 small cars at 50 miles per hour and records the braking distance. The sample average braking distance is computed as 98 feet. Assume that the population standard deviation is 20 feet. Calculate the value of the test statistic for null hypothesis H0: μ=90 when tested against alternative hypothesis Ha: μ≠90. Compare your test statistic obtained in previous question to critical value z alpha over two for alpha=0.01 and make conclusion based on your test.

My test statistic is greater than critical value z alpha over two, hence I reject the null hypothesis.

How do we choose which hypothesis is null and which is alternative?

Null hypothesis usually represents common knowledge, default situation, or a status-quo. Alternative hypothesis represents new knowledge.

A random sample is drawn from a some population with mean μ = 15 and standard deviation σ = 1.5. There are two samples, one of size n=20, another of size n=40. Can you conclude that the sampling distribution of the sample mean is normally distributed for both sample sizes?

Only sample mean with n=40 is approximately normal by central limit theorem since the sample size should be at least 30.

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X > 16) related to the probability P(X < 16)? (Hint: the best way to understand it is to draw normal distribution, denote mean, denote the events that we compare, and see how their probabilities are related).

P(X > 16) is greater than P(X < 16)

It is advertised that the average braking distance for a small car traveling at 50 miles per hour equals 90 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives 49 small cars at 50 miles per hour and records the braking distance. The sample average braking distance is computed as 98 feet. Assume that the population standard deviation is 20 feet. Calculate the value of the test statistic for null hypothesis H0: μ=90 when tested against alternative hypothesis Ha: μ≠90. Based on p-value for this test you found in previous question, make a decision whether to reject or not the null hypothesis. Use alpha=0.01 to make your decision.

Since p-value is smaller than alpha, I should reject the null hypothesis.

What kind of sampling is more likely to be used by political pollsters who want to ensure that each religious group is represented in the sample?

Stratified random sampling

The p-value is the likelihood of obtaining a sample mean that is at least as extreme as the one derived from the given sample, under the assumption that the null hypothesis is true.

True

The z table in our textbook provides cumulative probabilities.

True

True or false: For a given sample size n, a Type I error can only be reduced at the expense of a higher Type II error.

True

A random sample is drawn from a normally distributed population with mean μ = 15 and standard deviation σ = 1.5. There are two samples, one of size n=20, another of size n=40. Can you conclude that the sampling distribution of the sample mean is normally distributed for both sample sizes?

Yes, both the sample means will have a normal distribution.

A researcher in a small Midwestern town wants to estimate the mean weekday sleep time of its adult residents. He takes a random sample of 49 adult residents and records their weekday mean sleep time as 6 hours. Assume that the population standard deviation is known to be 2 hours. Calculate the 99% confidence interval for the population mean weekday sleep time of all adult residents of this Midwestern town. (Round final answers to 2 decimal places.)

[5.26, 6.74]

Due to symmetry, the probability that the standard normal random variable Z is greater than 0 is

equal to 0.5

Suppose you are constructing a confidence interval for the population mean. For a given confidence level and sample size, the width of the interval is wider for a

larger standard deviation