herbert stats PRACTICE TEST
Referring to Auditor, give the critical value(s). Equivalently, give the endpoint(s) of the rejection region.
+/- 3.182
The Humane Society reports that approximately 40% of all US households own at least one dog. In a random sample of 340 households, 153 households said that they owned at least one dog. Does this data provide sufficient evidence to indicate the Human Society was wrong using a 5% level of significance? We are attempting to show that the Humane Society is incorrect. Referring to Humane Society, what is the correct pair of hypotheses for this test?
H0: 𝑝 = 0.4 versus Ha: 𝑝 ≠ 0.4
A nutritionist believes that he can demonstrate that over 40% of adolescents in the United States are overweight. A 5% level of significance established. In order to test the nutritionist's hypothesis a simple random sample of 500 teenagers is obtained. 220 of the adolescents are found to be overweight. Referring to Nutritionist, what is the alternative hypothesis?
Ha : p>0.4
A political scientist believes that she can show that the average age of the voters in a presidential election is less than 40 years. To this end an exit poll of 100 voters is obtained. The data obtained in the poll has a sample average age of 39.2 years with a sample standard deviation of 8 years. We establish a 1% level of significance. Referring to Political Scientist, give the alternative hypothesis.
Ha : μ < 40
A credit manager at a chain for stores claims that the mean credit balance for the chain's customers is $105.5. An auditor suspects otherwise and believes he can show that this is false. To this end the auditor obtains a simple random sample of 4 balances of customer accounts. Assume the level of significance is 5%. The data are given below: Balance= 105, 95, 105, 95 Referring to Auditor, give the null and alternative hypotheses.
Ha : μ does not equal 105.5
A researcher wishes to show that an antibiotic will inhibit the growth of bacteria in a culture. The researcher inserts the antibiotic into 30 cultures and finds that 20 have reduced growth. A control group of 20 cultures is not given the antibiotic. Of these cultures, 10 have reduced growth. Test the hypothesis 𝐻0: 𝑝1 − 𝑝2 = 0 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1: 𝑝1 − 𝑝2 > 0 at a 5% level of significance. Here we let p1 denote the true reduced growth rate for the population of bacteria with the antibiotic, and let p2 denote the true reduced growth rate for the population of all bacteria without the antibiotic. Referring to Antibiotic give the pooled estimate of the hypothesized population proportion.
0.6
Suppose that we have events A and B so that P(A)=0.3 and P(B) = 0.4. Find P(A or B), if P(A|B)=0.2.
0.62
We wish to give a 95% confidence interval for the fraction of Louisiana adult residents who have a high school diploma or the equivalent. To this end we obtained a simple random sample of 300 Louisiana adult residents and found that 240 had a high school diploma. Referring to Diploma, what is the point estimate for the fraction of adult Louisiana residents that possess high school diplomas?
0.8
Referring to Diploma, give the 95% confidence interval for the true population proportion. (Leave the answer in the form: point estimate ± CV*SE)
0.8 =/- 1.96 * sqrt (0.8*0.2)/300
Suppose that the average appraised value of a house in Orleans Parish is $190,000 with a standard deviation of $20,000. If we obtain a simple random sample of 100 Orleans Parish houses, what is the probability that the sample average appraised value will fall between $186,000 and $193,000?
0.9104
Referring to Heights, suppose you take a random sample of 100 American women, and compute the mean height of the sample. What is the probability that the mean height will be at least 63.5 inches?
0.9772
A room contains three Urns: U1, U2, U3. U1 contains 3 red and 2 yellow marbles. U2 contains 3 red and 7 yellow marbles. U3 contains 1 red and 4 yellow marbles. Referring to Urns we enter the room and select an urn, but we are not sure which, and then we randomly remove a marble from the urn. Find the probability that the marble is red.
11/30
We wish to obtain a 90% confidence interval for the standard deviation of a normally distributed random variable. To accomplish this we obtain a simple random sample of 16 elements from the population on which the random variable is defined. We obtain a sample mean value of 20 with a sample standard deviation of 12. Give the 90% confidence interval (to the nearest integer) for the standard deviation of the random variable.
15 to 25
Referring to Auditor, calculate the test statistic assuming that 𝑠 = 5.5
2
Referring to Nutritionist, give the test statistic.
2
Suppose that 10% of Tulane students will remain in New Orleans throughout the summer. Suppose we randomly select 20 Tulane students. Referring to Tulane, find the expected number of students who will remain in New Orleans.
2
Referring to Independent Samples, what is the appropriate critical value for the 95% confidence interval?
2.042
Referring to VIT give the critical value
2.054
Referring to Random Variable, what is the value of the appropriate critical value?
2.576
Referring to Random Variable, compute the margin of error. (Leave the answer in the form of CV*SE)
2.576 * (3.4 / sqrt200 )
Suppose we have a continuous random variable Y with the probability density function 𝑓(𝑦) = { 1 2 (2 − 𝑦) 𝑓𝑜𝑟 0 ≤ 𝑦 ≤ 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is easy to see that the mean value of Y is given as 𝜇 = 2 3 . Calculate the variance of Y.
2/9
We wish to construct a 99% confidence interval for a random variable x. In order to accomplish this, we obtain a simple random sample of 200 elements from the population. We obtain a sample mean value of 20.5 and a sample standard deviation of 3.4. Referring to Random Variable, what is the point estimate of the true population mean value of x?
20.5
Referring to Random Variable, give the 99% confidence interval. (Leave the answer in the form of point estimate ± CV*SE)
20.5 +/- 2.576 * (3.4/sqrt200)
Referring to Position, if we randomly select 3 individuals for the position, what is the expected number of men?
21/11
Referring to Marbles, now suppose you select 4 marbles from the sack without replacement. That is, you select a marble, and you do not put it back in the sack before selecting the next marble. What is the probability of getting 3 blue ball and 1 red balls?
24/210
Referring to Position, if we randomly select 3 of the candidates from the group of 11, what is the probability that 2 men and one woman will be selected?
28/55
We wish to give a 95% confidence for the lead content of water in a Michigan county. To this end a simple random sample water from 4 locations gives the following concentrations in parts per million. Y=0, 5, 2, 5. Give the margin of error for the 95% confidence interval
3.182*√6/2
Referring to Auditor, give the point estimate of the true average credit card balance.
325
Referring to Auditor2, give the 90% confidence interval for the true average credit card balance. (Leave the answer in the form: point estimate ± CV*SE)
325 +/- 55.8
Referring to increase, find the critical value
4.541
Let X be uniformly distributed random variable in the interval [5,15]. Find the probability that X is at least 8 given that X is at most 13.
5/8
Referring to Auditor, give the margin of error for the confidence interval. (Leave the answer in the form: CV*SE)
55.8
Suppose you want to estimate the variance of a population whose mean value is µ and whose variance is 𝜎 2 . Taking a random sample of 15 observations of the population, you find that the sample mean value is 𝑦̅ = 4, and the sample standard deviation is s = 1.5. What are the appropriate critical values for the 90% confidence interval for the true population variance 𝜎 2
6.571 and 23.685
We wish to give a 95% confidence interval for μ1 − μ2. Here μ1 denotes the mean value of the population from which Sample 1 was obtained, and μ2 denotes the mean value of the population from which μ1 Sample 2 was obtained. Assume the populations are normally distributed and have equal, but unknown, variances. The results of the two samples are summarized in the table to the right. Referring to Independent Samples, compute the pooled estimate of the variance.
7
We have 7 men who are candidates for a position and 4 women who are candidates for the position Referring to Position, if we randomly select 3 of the 11 candidates for position, what is the probability that all three are men?
7/33
Referring to Final Exam, find a 90% confidence interval for students' exam scores, i. e., find and interval centered at 80 so that 90% of the exam scores fall within the interval.
80 +/- 8.225
Referring to Final Exam, find the upper 10% of exam scores.
86.4
Referring to Urns suppose that we enter the room and we select with replacement 5 marbles from U1 and 10 marbles from U2. Let Y denote a random variable defined as the total number of yellow marbles obtained from U1 and U2. Find the expected value of Y.
9
Referring to Humane Society, what is the conclusion of the test?
Fail to reject H0
We wish to demonstrate that taking vitamin C will shorten the average duration of cold symptoms by more than one day. To this end a random sample of 100 cold suffers is given a 100 milligram vitamin C supplement as long as the cold symptoms persist. We find a mean duration of 3.6 days with a standard deviation of 3 days. Independently, we observe the cold durations of a simple random sample of 50 cold sufferers who are given no supplement. We find a mean cold duration of 4.3 days with a sample standard deviation of 2√2 days. In what follows let µ1 denote the average cold duration with no supplement and let µ2 denote the average cold duration with the supplement. We establish a 2% level of significance. Referring to VIT C give the alternative hypothesis
Ha : μ1 - μ2 > 1
A producer of bakery items is testing a new leavening process used in producing bread. Loaves of bread were randomly sampled and analyzed for caloric content both before and after the new leavening process was used. The results of the two samples are given in the table below. Do these results give evidence to conclude that the number of calories has been reduced with the new leavening process? Use α = 0.05. Set μ1 denote the average number of calories per loaf with the new process and let μ2 denote the average number of calories per loaf with the old process. Referring to Calories, give the null and alternative hypotheses.
Ha: μ1 - μ2 < 0
What does the auditor assume of the credit card balances when developing this confidence interval?
balances are normal
Referring to Increase, find the test statistic.
𝟑√𝟑 /4
Referring to Yearning for Hypothesis Testing, find a 95% confidence interval for the true proportion of students studying statistics who are tired of confidence intervals and yearn for an understanding of hypothesis testing.
0.45 ± 1.96 ∗ √.45∗.55/200
An organization with a large automobile fleet would like the fleet to have an average gas mileage of more than 25 miles per gallon. In order to determine if this goal is being attained, they check the gasoline usage for 64 randomly selected company trips, finding a mean value of 26.8 mpg and a sample standard deviation of 4.8 mpg. We are testing the null hypothesis 𝐻0: 𝜇 = 25 versus the alternative hypothesis 𝐻𝑎: 𝜇 > 25 Referring to MPG compute the p-value.
0.0013
Suppose the average lifespan of a lab rat is 3.4 years with a standard deviation of 0.8 years. If we obtain a simple random sample of 64 lab rats, what is the probability that the sample average lifespan is at least 3.65 years?
0.0062
Referring to Nutritionist, give the p-value.
0.0228
Referring to Antibiotic give the test statistic
0.3333
A random sample of 200 students studying statistics contains exactly 90 students who are tired of confidence intervals and yearn to learn about hypothesis testing. Referring to Yearning for Hypothesis Testing, find the point estimate for the true population proportion of statistics students who tired of confidence intervals yearn to learn hypothesis testing.
0.45
Referring to Tulane, find the probability that at least 8 of the students in the sample will remain in New Orleans throughout the summer. You should use an n=20 table of cumulative binomial probabilities.
0.0004
A sociologist wants to show that the average family income of American families with two children is now more than $50,000. To this end the sociologist obtained the incomes of 900 families with two children. The sociologist will test the hypotheses H0: µ = 50,000 versus Ha: µ > 50,000 at a 5% level of significance. Suppose the sample of 900 family incomes has a mean value of 51,500 and a standard deviation of 15,000. Find the p-value.
0.0013
In a study to compare the effects of two pain relievers it was found that of 200 randomly selected individuals instructed to use Excedrin, 93% indicate that it relieved their pain, Of 450 randomly selected individuals instructed to use Tylenol, 96% indicated that it relieved their pain. We will develop a 90% confidence interval for the difference in the rates of pain relief for the two products. Let pop1 denote the population of headache sufferers who take Excedrin, and let Pop2 denote the population of headache sufferers who take Tylenol. Referring Pain Relief, what is the point estimate of the difference in the relief rates of the two pain relievers?
-.03
Referring to Pain Relief, give a 90% confidence interval for the true difference in the proportions that experienced pain relief. (Leave the answer in the form of point estimate ± CV*SE)
-0.03 +/- 1.645 * sqrt(0.93*0.07/200 + 0.96*0.04/450)
A research group wishes to assess the retention of information in purely visual ads versus audio visual ads. To this end an ad was prepared that presented information in a visual format only. This ad was presented to 100 consumers. A second ad presented the same information to 200 consumers, but the second ad presented the information in an audio- visual format. Each individual in each of the two groups was asked about 5 specific facts that were presented in each of the two ads. The visual only sample had an average of 2.68 correct responses with a standard deviation of 0.5. The audio- visual sample had an average of 3.1 correct responses with a sample standard deviation of 1. Let 𝜇1 denote the population average number of correct responses for the visual only group and let 𝜇2 for the audio- visual group. Give a 90% confidence interval for 𝜇1 − 𝜇2.
-0.42 +/- 1.645 * sqrt3/20
Referring to VIT give the test statistic.
-0.6
Referring to Calories, give the critical value(s)
-1.645
Calculate the critical value which would be used to test the hypotheses H0: μ = 4 versus Ha : μ < 4 at a 5% level of significance if we use a simple random sample of 12 elements. Assume the population standard deviation is unknown.
-1.796
A researcher wishes to determine if an antibiotic will inhibit the growth of bacteria in a culture. The researcher inserts the antibiotic into 30 cultures and finds that 20 have reduced growth. A control group of 40 cultures is not given the antibiotic. Of these cultures, 18 have reduced growth. Give a 99% confidence interval for the difference in the true rates of reduced growth for the cultures with the antibiotic and cultures without the antibiotic. Let p1 denote the true reduced growth rate for the population of bacteria with the antibiotic, and let p2 denote the true reduced growth rate for the population of all bacteria without the antibiotic.
.22 ±2.576*√ .67∗.33/ 30 + .45∗.55/40
A poll is taken by obtaining a simple random sample of 500 adults. It is found that 270 members of the sample believe more money should be spent on medical research. Give a 99% confidence interval for the true proportion of adults who believe more money should be spent on medical research.
.54 ± 2.576* √.54∗.46/500
Suppose the average value on the final exam is 80% with a standard deviation of 5%. Assume the exam scores are normally distributed. Referring to Final Exam, find the probability that a student scores between 72.5% and 85%
.7745
Let Y be a continuous random variable with probability density function 𝑓(𝑦) = { 3 32 (4 − 𝑦 2 ) 𝑓𝑜𝑟 − 2 ≤ 𝑦 ≤ 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Find the expected value of Y.
0
Referring to Calories, calculate the p-value.
0
Suppose that 50% of American adults have attended a college for at least one semester. If we select a simple random sample of 400 American adults, what is the probability that at least 300 of the individuals in the sample have attended a college for at least one semester?
0
The waiting time for the decay of a radioactive substance is exponentially distributed with a mean waiting time of 50 days. Find the probability that we wait for at most 30 days for the substance to decay.
1-e^-0.6
We wish to develop a 90% confidence interval for the difference in the time that cold sufferers experience symptoms without a vitamin C supplement and cold sufferers who take a vitamin C supplement. 55 individuals who reported cold symptoms were given 4 milligrams of vitamin C as long as the symptoms persisted. 55 individuals who experienced cold symptoms were given no vitamin supplement. The table below records the number of participants in each group, the mean duration of the cold symptoms and the standard deviation of the time that the cold symptoms persisted. Referring to Vitamin C, give the point estimate of the difference in the difference in the population mean values μ1 − μ2.
1.2
Referring to Vitamin C, give the 90% confidence interval. (Leave the answer in the form: point estimate ± CV*SE)
1.2 +/- 1.645 * sqrt(3.8^2/55 + 2.5^2/55)
Referring to Tulane, find the standard deviation of the number of students who will remain in New Orleans.
1.34
Referring to Antibiotic give the critical value
1.645
Referring to Ewoks and Wookies, if we wish to construct a confidence interval for the difference in true mean Ewok height and true mean Wookie height, what is the critical value(𝑧𝛼 2 ⁄ 𝑜𝑟 𝑡𝛼 2 ⁄ ,𝑛−1 ) that you should use for a confidence level of 90%?
1.645
Referring to Nutritionist, give the critical value(s). Equivalently, give the endpoint(s) of the rejection region.
1.645
Referring to Pain Relief, give the margin of error. (Leave the answer in the form of CV*SE)
1.645 * sqrt(0.93*0.07/200 + 0.96*0.04/450)
Referring to Vitamin C, give the margin of error for the 90%. (Leave the answer in the form: CV*SE)
1.645 * sqrt(3.8^2/55 + 2.5^2/55)
Referring to Humane Society, find the numerical value of the test statistic for the above measurements. Hint √ .4∗.6 340 = .03
1.667
Calculate z0.04. That is, compute the value so the area to the right of the value and beneath the standard normal curve is 0.04.
1.75
An auditor wants to give a 90% confidence interval for the average credit card balance for credit card holders of a chain of stores. To this end the auditor obtains a simple random sample of 9 balances of customer accounts. The data are given below: Balance= 410,455,390,175,340,340,200,340,270 It is easy to show that the sample average is 325 with the approximate standard deviation of 90 Referring to Auditor2, find the critical value that is appropriate for the 90% confidence interval.
1.860
Calculate t0.05,7. That is, compute the value so the area beneath the t-distribution of 7 degrees of freedom but to the right of the value is 0.05.
1.895
We wish to give a 90% confidence interval for the mean value of a normally distributed random variable. We obtain a simple random sample of 9 elements from the population on which the variable is defined. The sample has a mean value of 10.2 and a sample standard deviation 1.5. Find the 90% confidence interval for the mean value.
10.2 ± 0.930
Suppose that a researcher is investigating the hypotheses H0: µ=10 versus Ha: µ< 10 at a 5% level of significance. Suppose the researcher does accept µ< 10 when, in fact, µ< 10 is false. Then the researcher has made
A Type I error
Ralph is playing Deal or No Deal. He has a briefcase in front of him, and it contains one of three possible dollar amounts: $300,000, $75,000 or $ 300. Each possibility is equally likely. What is the expected value of Ralph's briefcase?
None of the above
Suppose that 30% of American like the taste of filbert nuts. Suppose you take a random sample of 100 Americans and calculate 𝑝̂, the sample proportion of people in your sample that like filbert nuts. Which of the following distributions does 𝑝̂most closely follow?
Normal with µ=0.3 and 𝜎 = 0.0458
Referring to Critical Values, what do we assume about the random variable x for which μ denotes the mean value?
X is normal
Referring to Calories, give the rejection rule and draw the rejection region
Z < -1.645
Referring to Political Scientist, describe the rejection region. Be sure to label the endpoint(s).
Z<-2.326
The height of adult females in the United States is normally distributed, with mean value 𝜇 = 64 and standard deviation 𝜎 = 2.5 inches. Referring to Heights, what portion of American women are between 69 and 70.25 inches tall?
a) 0.0166
Referring to Auditor, what must be true of the credit balances in order for the test to be valid?
balances are normal
Referring to Antibiotic what is the conclusion?
fail to reject H0
Referring to Political Scientist, what is the conclusion?
fail to reject H0
Referring to VIT what is the conclusion?
fail to reject H0
Referring to Auditor, should we reject the null hypothesis?
no
Referring to Tulane, should we use the normal approximation to the binomial (or the Central Limit Theorem for p̂) to approximate the probability computed in the preceding problem? Why or why not?
no
Referring to Calories, what is the conclusion?
reject H0
Referring to Nutritionist, what is the conclusion?
reject H0
Referring to Auditor, calculate the sample standard deviation of the data.
sqrt 33.3
Referring to Urns suppose that we enter the room and we select with replacement 5 marbles from U1 and 10 marbles from U2. Let Y denote a random variable defined as the total number of yellow marbles obtained from U1 and U2. Find the standard deviation of Y.
sqrt 33/10
Referring to Humane society, what is the rejection rule for this test?
z<-1.96 or z>1.96
A medical team is interested in pharmaceuticals that can raise the level of Factor X in the bloodstream. They obtain informed consent from 14 representative volunteers, and organize them into 7 pairs, each pair consisting of volunteers of the same sex, age and blood type. It is believed that age, sex and blood type may influence factor X. Within each pair one subject chosen at random is given the new experimental medication and the other subject is given a placebo. After 10 days the Factor X levels are measured with the following results: The medical team wishes to give a confidence interval for the average difference in Factor X for people who do not use Factor X (population 1) verses people who do take Factor X (population 2) The team wishes to develop a 90% for this difference. Let μ1 denote the average for the new medication and μ2 denote the average for the placebo. 76) Referring to Increase find the appropriate critical value for the 90% confidence interval
±1.943
Referring to Ewoks and Wookies, which of the following gives the 90% confidence interval for the true difference in mean Ewok height and mean Wookies height?
−1.1 ± 1.645 ∗ √3/30
Suppose you have a sack of 10 marbles, 4 of which are blue and 6 of which are red. Referring to Marbles, suppose you select 4 marbles from the sack with replacement. That is, you select a marble, record its color, and put it back into the sack. This is done a total of four times. What is the probability of getting 3 blue balls and 1 red ball?
𝐶43 * 4^3 * 6^1
The distance between mutations on a strand of DNA can be modeled using an exponential distribution. Suppose that the average distance between mutations is 1.6mm. Referring to Mutations, if you have identified one mutation, what is the probability the next mutation is less than 0.8 mm away?
𝑒^ −.5
You measure the height of n1=900 Ewoks, a species of teddy-bear-like hunter-gatherers that inhabit the forest moon of Endor. The average height you measure is 𝑦̅1 = 1 𝑚, and the standard deviation of the measurements is s1 = 1𝑚. Independently of the Ewok sample, you also measure the height of n2= 900 Wookies. The average height of the Wookies in the sample is 𝑦̅2 = 2.1𝑚 and the sample standard deviation is 𝑠2 = √2𝑚 In what follows population 1=Ewoks and Population 2 = Wookies Referring to Ewoks and Wookies, what is the unbiased point estimator for the difference in mean Ewok height and Wookie height?
𝑦̅1 − 𝑦̅2
