STA 2023 Practice Quiz 6

A parallel system of three components functions whenever at least one of its components works. Suppose that each component independently works with probability 0.40. What is the probability that the system is functioning?

0.784

Find a value z0 of the standard normal random variable Z such that p(Z <= z0) = 0.0401.

-1.75

In a test of H_0: mu = 100 against H_a: mu > 100, the sample data yielded the test statistic z = 2.17. Find the p-value for the test.

0.015

If x is a binomial random variable with n = 25, and p = 0.8. Use binomial table to find p(x > 20).

0.421

Let A and B be two events defined on a sample space S of an experiment such that p(A union B) = 0.8, and p(B) = 0.3. What is the probability of A if events A and B are disjoint events?

0.5

Let A and B be two subsets of the sample space of an experiment. If P(A) = 0.4, P(B) = 0.5, and P(A intersection B) = 0.1, find P(A union (Bc)).

0.6

Find the area under the standard normal distribution between -2.05 and -1.59.

0.9239

For the standard normal random variable Z, find p( Z > - 2.33)

0.9901

Find a value z0 of the standard normal random variable Z such that p(-1.5 < Z <= z0) = 0.7793

1.02

The following table shows the distribution of 40 students by number of credit cards. number of credit cards Number of Students (x) (f) 0 6 1 20 2 10 3 4 Find the mean number of credit cards.

1.3

Find a value z0 of the standard normal random variable Z such that p( Z >= z0) =0.05

1.645

The speed of the fastball thrown by 120 major league baseball pitchers was measured by radar gun. The average fastball was thrown at 85 miles per hour (mph). The standard deviation of the speeds was 5 mph. Which of the following fastball speeds would be classified as outliers?

101

You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years of ownership. Since you are particularly interested in Toyota Camry, you decide to estimate the resale value of the Camry with a 95% confidence interval. You manage to obtain data on 25 recently resold 5 year old Toyota Camrys. These 25 cars were resold at an average price of $12,400 with a standard deviation of $700. Assuming that the sampled population is normal, find a 95% confidence interval for the true mean resale value of a 5 year old Toyota Camry.

12111.04, 12688.96)

Find the minimum number of cellular phones that a manufacturer must test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of 0.10 is used for p?

2436

In a promotion at a store, each customer gets a chance to randomly draw a ticket from a box. There are 100 tickets. 20 tickets say "Winner!" and 80 tickets say "Sorry. Try again next time." Assume two customers play and that ticket is NOT replaced after each customer plays. What is the probability when two customers play that both win?

38/990

The FCAT math scores of Florida high school students is normally distributed with mean μ = 77 and standard deviation σ= 7. Which of the following statements is NOT correct?

All (exactly 100%) Florida high school students scored between 56 and 98

In general, large p-values support the alternative or research hypothesis.

False

Which of the followings is a measure of position?

Median

Which of the following statements is true?

Median, Percentiles and Quartiles are measures of Position

Specify the rejection region associated with the test of H_0: mu = 10, H_a: mu > 10 when alpha = 0.05, and n = 121.

Z > 1.645

For any two independent events, which of the followings is correct ?

p(A/B) = p(A) A or B is the same as A union B. Thus the addition law says p(A or B) = p(A) + p(B) - p(A intersection B). Note however that p(A intersection B) = p(A). p(B) since A and B are independent. So if you replace "or" by "intersection" in the left side, then it would be a true statement.

To approximate binomial probability p(x > 8) when n is large, identify the appropriate 0.5 adjusted formula for normal approximation.

p(x > 8.5)

A random sample of 400 satellite radio subscribers were asked "Do you have a satellite radio receiver in your car?". The survey found that 280 subscribers did, in fact, have satellite receiver in their car. Find a 90% confidence interval for the true proportion of satellite radio subscribers who have a satellite radio receiver in their car.

(0.662, 0.738)

Consider the pharmaceutical company that desire an estimate of the mean increase in blood pressure of patients who take a new drug. The blood pressure increases (measured in points) for n = 6 patients in the human testing phase are found as 1.7, 3.0, 0.8, 3.4, 2.7, 2.1. The mean and variance of these 6 values are 2.283 and 0.902. Find a 99% confidence interval for the true mean increase in the blood pressure associated with the new drug for all patients in the population. Assume that the blood pressure data follows a normal distribution.

(0.72, 3.85)

Consider a large hospital that wants to estimate the average length of stay of its patients. The hospital randomly samples n = 100 of its patients and finds that the sample mean length of stay is 4.5 days. Assume that the standard deviation of the length of stay for all hospital patients is 4 days. Find a 95% confident interval for true mean length of all hospital patients.

(3.72, 5.28) You need to use the large sample confidence interval formula for mean. The z-score for 95% confidence in 1.96. So the lower limit of the confidence interval is 4.5 - (1.96)(4)/10 = 3.716. Similarly the upper limit is 4.5 + (1.96)(4)/10 = 5.284. Thus a 95% confidence interval is (3.716, 5.284).

The FNE ("fear of negative evaluation") scores of bulimic students have a distribution that is normal with mean =18 and standard deviation = 5. Consider a random sample of 49 students with bulimia. What is the probability that the sample mean FNE score is less than 18.5?

0..7580

For the standard normal random variable Z, find p( Z < - 2.08)

0.0188

The probability that a student is accepted to a prestigious college is 0.5. Assume that this probability is the same for each of a group of 100 independent students who applied for admission. What is the approximate probability that at least sixty will be accepted?

0.0287

In a test of H_0: mu = 100 against H_a: mu < > 100, the sample data yielded the test statistic z = 2.17. Find the p-value for the test.

0.03

In a promotion at a store, each customer gets a chance to randomly draw a ticket from a box. There are 100 tickets; 20 tickets say "Winner!" and 80 tickets say "Sorry. Try again next time." Assume two customers play and that the ticket is replaced after each customer plays. What is the probability when two customers play that both win?

0.04

For the standard normal random variable Z, find p( 1.09 < Z < 4.64)

0.1379. It is a good practice to draw the bell curve with 0 marked in the middle on the horizontal axis. Then mark 1.09 and 4.64 on the horizontal line, color the area between 1.09 and 4.64. This colored area is your answer. To find this area, you need to subtract the area between 0 and 1.09 from the area between 0 and 4.64. Look carefully in your table IV, there is no number more than 3.09; the area between 0 and 3.09 is 0.499. How do we find the area between 0 and 4.64? Since the table does not go over 3.09, we need to approximate it. Since the entire area to the right of zero is o.5 and the area between 0 and 3.09 is approximately 0.499, we can say that the area between 0 and 4.64 is more than 0.499 but less than 0.5. Instead of choosing a number between 0.499 and 0.5, we simply take 0.5 as the approximate area between 0 and 4.64 (NOTE: the area between 0 and any number "k" is always approximated by 0.5 whenever "k" is greater than 3.09). So the area between 1.09 and 4.64 is equal to (0.5 - area between 0 and 1.09) which is 0.5 - 0.3621 = 0.1379.

Scores (X) on a college entrance examination are normally distributed with µ=540 and σ=100. If you select one student at random, what is the probability that the selected student will have a score greater than 640?

0.1587

In a certain city where it rains frequently, records have been kept and relative frequencies have been used to estimate these probabilities. The probability that rain is predicted on a day is 0.2. The probability that it actually rains on a day that rain is predicted is 0.9. The probability that it actually rains on a day that rain is not predicted is 0.3. For a randomly selected day, what is the probability that rain is predicted and it does rain?

0.18

If x is a binomial random variable with n = 20, and p = 0.2. Use binomial table to find p(x = 4).

0.219

The FNE ("fear of negative evaluation") scores of bulimic students have a distribution that is normal with mean =18 and standard deviation = 5. Consider a random sample of 49 students with bulimia. What is the probability that the sample mean FNE score is less than 17.5?

0.242 You need to find the probability that the mean FNE score of 49 students is less than 17.5, i.e. you want p( x-bar 17.5). You need z-score for 17.5 using the Z formula involving x-bar since you are dealing with x-bar. The z-score for 17.5 is - 0.7. Now find p( Z - 0.7) using z-table. This is simply the entire area to the left of -0.7 which is equal to (0.5 - area between 0 and 0.7). The answer is then 0.5 - 0.2580 = 0.242.

In a study of Emergency Medical Services (EMS) ability to meet the demand for an ambulance, a researcher presented the following scenario. An ambulance station has one vehicle and two demand stations, A and B. The probability that the ambulance can travel to a location in under eight minutes is 0.58 for location A and 0.42 for location B. The probability that the ambulance is busy at any given point in time is 0.3. Find the probability that the EMS can meet demand for an ambulance at location B .

0.294

Using the table below showing the probability distribution for winning a prize in a game of chance, find the probability of truly winning money that is more than $0.0. X(amount won) $0 $1 $10 $100 p(x) .70 .25 .04 .01

0.30 You may look at two ways. Find the probability of winning $0 which is 0.70 and then subtract it from 1 to get the correct answer 0.30. Or find the probability of winning more than zero dollars (in this case $1 with probability = 0.25, $10 with probability 0.04, and $100 with probability 0.01) by adding all three probabilities of winning to get the correct answer 0.25 + 0.04 + 0.01 = 0.30.

A binomial experiment has 10 trials with probability of success 0.20 on each trial. What is the probability of less than two successes?

0.376

The Statistical Abstract of the United States reports that 30% of the country's households are composed of one person. If 20 randomly selected homes are to participate in a Nielson survey to determine television ratings, find the probability that fewer than six of these homes are one-person households.

0.416

The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of 2 ounces. Suppose 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags falls between 10.20 and 10.50 ounces.

0.4332

The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of 2 ounces. Suppose 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags falls between 10.50 and 10.80 ounces.

0.4332

To gauge their fear of going to a dentist, a large group of adults completed the Modified Dental Anxiety Scale questionnaire. Scores (X) on the scale ranges from zero (no anxiety) to 25 (extreme anxiety). Assume that the distribution of scores is normal with mean µ= 11 and standard deviation σ= 4. Find the probability that a randomly selected adult scores between 10 and 15.

0.44 We want p(10 X 15). Find the z-scores for both 10 and 15. These are (10-11)/4 = -0.25 and (15-11)/4 =1. Thus p(10 X 15) = p(-0.25 Z 1). Now draw the bell curve with zero in the middle and -0.25 to the left and 1 to the right of zero. Your answer is the area between -0.25 and 1. Using chart, the area between 0 and -0.25 is 0.0987, and the area between 0 and 1 is 0.3413. The total area is then equal to 0.0987 + 0.3413 = 0.44

In a certain city where it rains frequently, records have been kept and relative frequencies have been used to estimate these probabilities. The probability that rain is predicted on a day is 0.2. The probability that it actually rains on a day that rain is predicted is 0.9. The probability that it actually rains on a day that rain is not predicted is 0.3. For a randomly selected day, what is the probability that the prediction is correct?

0.74

For the standard normal random variable Z, find p( - 2.09 < Z < 1.64)

0.9312

Fifty percent of all drivers wear their seat belts. A random sample of n=100 drivers has been taken. Find the probability that fewer than 60 were wearing their seat belts?

0.9713 You need Binomial p(X 60) when n = 100, p(success) = 0.5. Since n is large, use normal approximation for Binomial probability. Need μ = np = 50 ; σ2 = np(1-p) = 25; so σ = 5. Now do 0.5 adjustment; Binomial p(X 60) = normal p(X 59.5) = p(z 1.9) which is the entire area to the left of 1.9. This area is equal to 0.5 + 0.4713 = 0.9713.

A human gene carries a certain disease from the mother to the child with a probability rate of 70%. Suppose a female carrier of the gene has three children. Also assume that the infections of the three children are independent of one another. Find the probability that at least one child gets the disease from their mother.

0.973

The SAT scores (x) of Florida high school students are normally distributed with mean µ= 1200 and standard deviation σ= 100. Top 33% of these students are expected to get full tuition scholarship. What is the minimum score for this scholarship?

1244

The Northeast Home Builders Association conducted a research project to estimate the average number of days it took to construct a new home. They decided to randomly sample completed new homes and collect the total number of days needed to construct the homes. How many homes (at a minimum) should they sample to estimate the true mean number of days to within 10 days with 95% confidence. Assume that the number of days for all completed homes range from 90 to 350.

163

Find a value z0 of the standard normal random variable Z such that p(1 < Z < z0) = 0.1359

2.0 You need to add the area between 0 and 1 to 0.1359 to identify the area between 0 and z0 and then find the corresponding z-score. See the explanation below for correct answer: First you need to guess correctly about the position of z0 in relation to zero and 1 in your bell curve (draw the bell curve and mark zero and 1 and then z0 on the horizontal line). Your z0 must be to the right hand side of 1.0 on the horizontal line. Now to find z0, you need to know the area between 0 and z0 which can be obtained by adding 0.3413 (the area between 0 and 1) to 0.1359 (the given area between 1 and z0). This total area is 0.4772 which is the area between 0 and z0. Now look at the chart area for 0.4772 (or the value closest to 0.4772). We have 0.4772 corresponding to z0 = 2 which is the answer.

Psychologists tend to believe that there is a relationship between aggressiveness and order of birth. To test this belief, a psychologist chose 500 elementary school students at random and administered each a test designed to measure the student's aggressiveness. Each student was classified according to one of four categories. The number of students falling in the four categories are shown here. Firstborn Not Firstborn Aggressive 60 90 Not Aggressive 140 210 A student selected at random from these 500 students is found to be aggressive. What is the probability that the student is firstborn?

2/5

In five recent weeks, a town reported 36, 29, 42, 25, and 28 burglaries. Find the median number of burglaries for these weeks.

29

Consider the probability distribution shown for the random variable x here: x: 1 2 4 10 p(x): 0.2 0.4 0.2 0.2 Find the expected value (i.e., mean value) of x.

3.8

A psychologist has developed a new test of spatial perception, and she wants to estimate the mean score achieved by adult male pilots. Find the minimum number of people that must be tested if she wants to estimate the true mean with an error of no more than two points with 90% confidence. An earlier study suggests that the population standard deviation sigma is equal to 21.2.

305 Use the sample size formula for the purpose of estimating true mean. Here z-score for the given 90% confidence is 1.645, sigma is equal to 21.2 and SE = 2. Thus n = (1.645)(1.645)(21.2)(21.2)/(2*2) = 304.05. Since the sample size cannot be fractional number, you need to choose 305 or more. So the minimum n is 305.

The following table summarizes the race and positions of 368 National Basketball Association (NBA) players in 1993. Guard Forward Center Total White 26 30 28 84 African-American 128 122 34 284 Total 154 152 62 368 What proportion of players is African-American or Center Players?95

312/368

In a promotion at a store, each customer gets a chance to randomly draw a ticket from a box. There are 100 tickets. 20 tickets say "Winner!" and 80 tickets say "Sorry. Try again next time." Assume two customers play and that the ticket is NOT replaced after each customer plays. What is the probability when two customers play that exactly one customer wins and the other loses?

32/99 you want the probability of one customer wins and the other loses. Here (i) above gives you the probability that only first customer wins and second customer loses, and (ii) above gives the probability that first customer loses and second customer wins. You need to add the two numbers in (i) and (ii) to get the correct answer 16/99 + 16/99 = 32/99 since these two are two disjoint options for the event in the question you are trying to answer.

How many students (minimum possible) should be sampled if you want to estimate the true mean number of credit hours per student with an error of no more than 0.3 and 95% confidence. From a prior study, it is known that the standard deviation is 2.8.

335

Suppose that the scores (X) on a college entrance examination are normally distributed with a mean score of 560 and a standard deviation of 40. A certain university will consider for admission only those applicants whose scores fall among the top 67% of the distribution of scores. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university.

542.40

A 95% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (3%, 9%). What is the point estimator of the mean percentage of reservations that are canceled on the day of the flight?

6%

A consumer testing service is commissioned to pick top three brands of laundry detergent from ten brands of which three brands are from company X and the remaining seven are from company Y. Assume that the testing service makes its choice by a random selection. What is the probability that exactly two of company X's brands is selected in the top three?

7/40

The age of patients in an adult care facility averages 75 years and has a standard deviation of five years. Assume that the distribution of age is bell-shaped symmetric. Find the 16th percentile in the age distribution.

70 years

In a promotion at a store, each customer gets a chance to randomly draw a ticket from a box. There are 100 tickets. 20 tickets say "Winner!" and 80 tickets say "Sorry. Try again next time." Assume two customers play and that the ticket is NOT replaced after each customer plays. What is the probability that the second customer loses, given that the first customer wins?

80/99

n a promotion at a store, each customer gets a chance to randomly draw a ticket from a box. There are 100 tickets. 20 tickets say "Winner!" and 80 tickets say "Sorry. Try again next time." Assume two customers play and that the ticket is NOT replaced after each customer plays. What is the probability that the second customer loses, given that the first customer wins?

80/99

If nothing is known about the shape of the distribution of a large dataset, what percentage of data fall within 2 standard deviation of the mean?

At least 75%.

A long-range missile missed its target by an average of 0.88 miles. A new steering device is supposed to increase accuracy, and a random sample of 8 missiles were equipped with this new mechanism and tested. These 8 missiles missed by distances with a mean of 0.76 miles and a standard deviation of 0.04 miles. Suppose that you want the probability of Type I error to be 0.01. State the research hypothesis to answer the question " Does the new steering system lower the miss distance?" Assume that the sampled population is normal.

H_a: mu < 0.88 (i.e., true mean missed distance for all missiles is less than 0.88)

Which is a measure of dispersion?

Range

Which of the following statements is False?

The median of the dataset 1, 4, 6, 5, 8 is equal to 6.

For a data distribution that is skewed to the left, Mean < Median.

True

The distribution of scores of 300 students on an easy test is expected to be skewed to the left.

True

The smaller the p-value associated with a test of hypothesis, the stronger the support for the research hypothesis.

True

True or False. For a specified sampling error (SE), increase in the confidence level (1-alpha) will lead to a larger n in determining the sample size

True

Which of the followings is not included in five-number summary results?

Variance

A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. A typical test performed by the FDA is the following: The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table below . Let mu_1 be the true mean weight of individuals before starting the diet and let mu_2 be the true mean weight of individuals after 3 weeks on the diet. Person Weight Before Diet (x_1) Weight After Diet (x_2) 1 161 154 2 206 201 3 199 196 4 208 202 5 215 211 FDA wants to determine if the diet is effective at reducing weight. What is their research hypothesis? (Assume that the difference of weights follow normal distribution)

mu_1 - mu_2 > 0

researcher is interested in comparing two teaching methods for slow learners. In particular, the researcher wants to determine if a new method of teaching is better (gives higher scores) than the standard method currently used. Type I error rate is set at alpha= 0.05. Ten slow learners are taught by the new method and 12 by the standard method. The results of a test at the end of the semester are given below (assume that the normal distribution with equal variances assumptions are satisfied) Test scores (new method): 80, 76, 70, 80, 66, 85, 79, 71, 81, 76. Test scores (standard method): 79, 73, 72, 62, 76, 68, 70, 86, 75, 68, 73, 66. What is the appropriate research hypothesis? (assume mu_1 = true mean of all scores under new method, and mu_2 = true mean of all scores under standard method)

mu_1 - mu_2 > 0

A long-range missile missed its target by an average of 0.88 miles. A new steering device is supposed to increase accuracy, and a random sample of 8 missiles were equipped with this new mechanism and tested. These 8 missiles missed by distances with a me an of 0.76 miles and a standard deviation of 0.04 miles. Suppose that you want the probability of Type I error to be 0.01. If you are asked to do a test to address the question "Does the new steering system lower the miss distance?" , what would be the appropriate rejection region associated with your test? Assume that the sampled population is normal.

t < -2.998 You do a one-tailed t-test with rejection region to the left of the t-distribution with df = n-1 = 7. The t-value from t-table corresponding to row 7 and column t_.01 (since alpha = 0.01) is 2.998 and you insert the negative sign since your test is one-tailed with research hypothesis H_a: mu 0.88.

A researcher is interested in comparing two teaching methods for slow learners. In particular, the researcher wants to determine if a new method of teaching is better (gives higher scores) than the standard method currently used. Type I error rate is set at alpha= 0.05. Ten slow learners are taught by the new method and 12 by the standard method. The results of a test at the end of the semester are given below (assume that the normal distribution with equal variances assumptions are satisfied). Test scores (new method): 80, 76, 70, 80, 66, 85, 79, 71, 81, 76. Test scores (standard method): 79, 73, 72, 62, 76, 68, 70, 86, 75, 68, 73, 66. The researcher stated the research hypothesis correctly as H_a: mu_1 - mu_2 > 0. What is the appropriate rejection region? Here mu_1 = true mean of all scores under new method, and mu_2 = true mean of all scores under standard method.

t > 1.725

A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. A typical test performed by the FDA is the following: The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table below . Let mu_1 be the true mean weight of individuals before starting the diet and let mu_2 be the true mean weight of individuals after 3 weeks on the diet. Person Weight Before Diet (x_1) Weight After Diet (x_2) 1 161 154 2 206 201 3 199 196 4 208 202 5 215 211 FDA wants to conduct a test to determine if the diet is effective at reducing weight using alpha = 0.05. What is the rejection region of their test? (Assume that the difference of weights follow normal distribution)

t > 2.132. You have paired data and you need to use the paired t-test with DF =n-1 = 4. Your test is one-tailed to the right and so the rejection region lies to the right side of the t-distribution. Look at the row 4 and column t_0.05 in the t-table and you get 2.132. The rejection region is t > 2.132.

For a fixed confidence level (1-alpha), increasing the sampling error (SE) will lead to a smaller n in determining the sample size.

true