MATH 110 EXAM II Past Exam Questions
1)Suppose 27% of Mt.SAC students are vocational students. When 100 Mt. SAC students are chosen randomly, find use normal approximation to the binomial to compute the probability of selecting 30 or fewer vocational students.
2)A) Give an example of a continuous variable that is not normal. Justify your answer. B) Give an example of a discrete random variable that is not binomial.
3) Credit card balances are normally distributed with a population mean of $2870 and a population standard deviation of $900. You must first define x
a)What is the probability that the mean balance of 40 credit card holders is between $2850 and $2900?
b)What is the probability that a randomly chosen credit card holder has a balance less than $2840?
4)IQ scores are normally distributed with a mean 100 and a standard deviation 16. Define x and find the 97th percentile IQ score.
5)The director of admissions at a university wants to estimate the mean and standard deviation of the family incomes of their freshmen. A sample of 6 freshmen produced the following results:
21,000 31,000 23,000 43,000 54,000 30,000
Assume that incomes are normally distributed.A)Construct a 99% confidence interval for the mean family income. You must find the point estimate and the margin of error. B) Construct a 99% confidence interval for the population standard deviation. .
6)How large a random sample is required to estimate the population proportion of whatever with 95% confidence to within 5 percentage points?
7)It is estimated that 20% of Latinos aged 45-65 have diabetes. In a random sample of 100 Latinos, construct the unusual region to determine if it is unusual for 80 people to have diabetes. You must first define x.
8)The wait times to be seated for dinner at a restaurant are normally distributed with a mean 7.2 minutes and a standard deviation of 1.5 minutes. Find the probability that the mean wait time of 20 randomly selected patrons is more than 8 minutes. You must first define x.
9)When 500 college students are randomly selected and surveyed, it is found that 24% of them own notebook computers. Construct a 98% confidence interval to estimate the true percentage of college students who own notebook computers. You must define the parameter to be estimated.
10)An automotive battery manufacture claims that the mean life of their batteries is more than 60 months. A sample of 35 batteries had a mean life of 58.12 months and a standard deviation of 3.2 months. Construct a 98% confidence interval for the population mean. You must first define the parameter to be estimated.
11)Weights of new born babies in the United States are normally distributed with a mean of 3500 grams and a standard deviation of 500 grams. You must first define x
a)Find the probability that the weight of a randomly selected baby is between 3600 grams and 3800 grams.
b)Find the weight that separating the top 5% from the others.
12) A survey of Internet users found that 80% favored government regulation of junk email. When 200 people are chosen at random, construct the “unusual” region for the number of people who favor government regulation on junk mail.Also determine whether it is unusual to have 180people in favor of the junk mail regulation? You must first define x
13)A union official wanted an idea of whether a majority of workers at a large company would favor a contact proposal. She surveyed 491 workers and found that 254 favored the proposal. Find a 97% confidence interval for the populationproportion of all the workers who favor the proposal. You must define the parameter to be estimated.
14)A town official wants to estimate the population proportion of voters favoring a nuclear freeze in the city. How large a random sample is required to estimate the population proportion within four percentage points with 99% confidence? You must define the parameter to be estimated.
15)The following data represent the IQ scores of 11 randomly selected Mt. SAC students.
11090 110 120 96 110 165 110 100 91 112
a)State the assumption necessary to construct a valid confidence interval for the population standard deviation of IQ scores of Mt. SAC students.
b) Assuming a), construct a 95% confidence interval a) for the population mean b) for the population standard deviation of IQ scores of Mt. SAC students. You must define the parameter to be estimated.
16)The HDL cholesterol of females is normally distributed with a mean 53 and a standard deviation 13.4. What is the probability that a random sample of 20 females will have mean HDL cholesterol above 56? You must first define x.
17)The drug Lipitor is meant to lower cholesterol levels. In a clinical trial of 820 patients, 51 reported headache as a side effect. Construct a 94% confidence interval for the population proportion of Lipitor users who will report headache as a side effect. You must define the parameter to be estimated.
18) According to a recent study, 40% of teachers in California have considered changing their careers. Use normal approximation to binomial, find the probability that in a random sample of 300 teachers, a) 90 or more teachers have considered changing their careers b) lessthan or equal to 100 teachers have considered changing their careers c) exactly 110 teachers have considered changing their careers.
19)A sociologist wishes to estimate the population proportion of residence in Los Angeles living in poverty. How large must the sample be if she wishes the estimate to be within 2 percentage points with 96% confidence. You must define the parameter to be estimated.
20)The Department Of Transportation is investigating the population standard deviation of commuting distances for the Los Angeles residents. A random sample of eight commuting distances (in miles) is shown below. Construct a 90% confidence interval for the population standard deviation of commuting distances. You must first define the parameter to be estimated.
12 19 10 8 13 20 21 33
21)According to a survey, 48% of American workers think that they will be working when they are 65 or older. Using the normal approximation to the binomial, find the probability that in a random sample of 200 American workers, a) 80 to 90 will say that they will still be working when they are 65 or older? B) 100 or more will say that they will still be working when they are 65 or older? You must first define x.
22)a. Find the probability of guessing at least 4 out of 5 multiple-choice questions correctly when each question has four possible answers.
b. It is estimated that 20% of Latinos aged 45-65 have diabetes. In a random sample of 10 Latinos, find the probability that at least three will have diabetes.
23)A survey of Internet users found that 80% favored government regulation of junk email. When 20 people are chosen at random, find the probability that at least 18 people are in favor of government regulation of junk email. Also find the mean and standard deviation.
24)Find the expected value for the following gamble: Two cards are drawn from a deck of 52 cards without replacement. If two hearts are drawn, you win $16. Otherwise, you lose $4.
25)A coin is tossed 3 times and let x denote the number of heads observed. Construct a probability distribution table and use the table to compute the expected value and population standard deviation.
26)
a)A normal distribution has a ______-______curve that satisfies 68-95-99.7 rule.
b)The standard normal distribution is a normal distribution with
c)If the distribution of x is ______, the distribution of is normal regardless of the sample size.
d)(True/false) A normal random variable is continuous.
e)True/false A 95% confidence level indicates the probability that the confidence interval contains the actual parameter is 95%.
f)The margin of error represents ______error. The errors occur because of human mistakes are called ______errors.
g)If a test consists of 10 multiple choice questions, the mean number of correct guesses is ______if each question has 4 possible answer
27)The data below represent adult single-day ski lift tickets prices (in dollar) of 10 randomly chosen ski resorts in California. Construct a 90% confidence interval for the standard deviation of single-day ski lift tickets prices. You must first define the parameter to be estimated and compute the margin of error.
90 80 70 85 95 70 80 90 95 100
28)The probability of winning on a slot machine is 5%. If a person plays the machine 300 times, use normal approximation to binomial to compute the probability of winning less than or equal to 20 times.
29)A life insurance company sells one-year $20,000 policy for $4,000. Compute the expected value of the policy if probability that the person survives the year is 0.9.
30)It is reported that 77% of workers drive to work alone. When 10 workers are chosen randomly, find the probability that at least two drive to work alone. You must first define x. In addition, construct the “unusual “ region.
31)A person pays $3 to play a certain game by rolling a die. If the player rolls a 5, he or she wins $12. Otherwise, the person wins nothing. Find the expected net value for this game.
32)) In Mt. SAC, 90% of the incoming first-year students enroll in a remedial math course. Use normal approximation to binomial to find the probability that of 100 randomly selected incoming freshmen, 95 or more enroll in a remedial math course. First define x.
33)To qualify for security officers’ training, recruits are tested for stress tolerance. The scores are normally distributed with a mean of 6.2 and a standard deviation of 1.4. If only the top 13% of recruits are selected, find the cutoff score.
34)A recent study showed 40% of college students work full-time. When 100 college students are chosen randomly, use normal approximation to binomial to find the probability that 50 or fewer college students work full-time .
35)The time it takes a group of adults to complete a certain exam is normally distributed with a mean 46.2 minutes and a standard deviation 4.2 minutes. Find the probability that the mean time to complete the exam for 30 randomly chosen adults will be less than 45 minutes. You must first define x.
36)
a)True/false: The expected value of a discrete random variable can be thought as a long-term average.
b)True/false: In a binomial experiment, the outcomes must be dependent.
c)True/ false: is a parameter.
37)A professor wants to estimate the mean length of time that a student must wait to see him after arriving at the office for his office hours. A random sample of 8 waiting times in minutes is shown below. Find a 99% confidence interval for the population mean waiting time. You must first define the parameter to be estimated.
10 19 20 21 21 20 40 30
38)entering the intersection of Grand and Temple.
3443 48 53 49 36 45 51 41 49
Construct a 95% confidence interval for the population mean speed of all cars entering the intersection of Grand and Temple. You must first define the parameter to be estimated.
39)The times that Mt.SAC students spend studying per week are normally distributed with a mean of 6.4 hours and a standard deviation of 2 hours. Define x and find the time that separates the top 4% from the rest.
40)The CDC reported that 20% of preschool children lack required immunizations. When 10 preschool children are chosen randomly to find the probability that at least two children lack required immunizations
41)A survey of 500 randomly selected Mt. SAC students showed that 90 of them were Baptists. Construct a 98% confidence interval for the population proportion of Mt. SAC students who are Baptists. You must first define the parameter to be estimated.
42)The life of a calculator is normally distributed with a mean of 440 hours and a standard deviation of 120 hours. When 30 calculators are chosen randomly, find the probability that their mean life is less than 440 hours. You must first define x.
Solutions (again, the answers may not be correct )
1) Apply continuity correction.
. Compute the mean and std. dev.. Now get the z-score: . Invnorm(-10,0.80,0,1)=.7881
2)a) x = Exact age of death. The distribution is skewed to the left. B) x = outcome when a die is tossed. More two possible outcomes.
3) x = credit card balance of a person
a) (use Normalcdf(-.14, .21, 0, 1))
b)
(use normalcdf(-1000, -0.03, 0, 1))
4) x = IQ score of a person First find the s-score: Invnorm (0.97, 0, 1)=1.88 Now find x using x=100+15(1.88) = 130.08
5) a) mu = population mean family income of all freshmen:,
invt(0.995. 5)= 4.032
First store the values in L1: Stat<Edit
Stat>Calc<One Var Stats
invt(0.005. 6)= 4.032,
b) sigma=population standard deviation of family incomes of all freshmen First compute s using TI. stat>calc>edit to store the values, then stat>calc>one var stats: . Next compute . Then go to the chi- square table to get before applying the formula.
6) Here find the sample size (key words?) First compute . Then invnorm (.975, 0, 1)=1.96 . Now apply the formula.
7) x = of Latinos in the sample having diabetes unusual region is
So 80 is in the unusual region.
8)X is the waittime to be seated. You need to use since you are asked to find the probability of mean..
. Invnorm(2.39, 10, 0, 1)=0.08
9) p = population proportionof studentswho own laptops: Then invnorm (.99, 0, 1)=2.33 .
10) mu = population mean life of a battery: , df=35-1=34. InvT(0.99,34)=2.441 ,
11) x= weight of a new born babya)
(Use normalcdf(0.2, 0.6, 0, 1) b) Here you are asked to find x. First find the area on the left side by 1-0.05 = 0.95. Then the z-score is invnorm(0.95, 0, 1)=1.64. . Now find x: grams.
12) # of people in the sample in favor of government regulation of junk email unusual region is
So 180 is in the unusual region.
13) p = population proportion of workers who favor a contact proposal. Then invnorm (.985, 0, 1)=2.17 .
14) p = population proportion of voters favoring a nuclear freeze in the city. Here you are asked to find the sample size. Then invnorm (.995, 0, 1)=2.58 . 1041people (always round up to a whole number)
15) a) The distribution of the IQ scores of Mt SAC students must be normal. B) a) mu-mean IQ score of Mt. SAC students. Compute the sample mean and sample std.dev using TI; stat>calc>edit to store the values, then stat>calc>one var stats:
Df=11-1=10, invT(1-0.025,10)=2.228
b)
16) x = HDL level of a female
(use Normalcdf(1.00, 10000, 0, 1))
17) p = population proportion of Liptor users who get headaches. Then invnorm (.97, 0, 1)=1.88
18) x is the number of teachers have considered changing their careers. First find the mean and standard deviation using the binomial formulas: . Then apply normal correction a) . Normalcdf(-3.56, 1000, 0, 1) b) . Normalcdf(-100, -2.29, 0,1) c) . Normalcdf(-1.24 -1.12, 0, 1) )
19)
p = population proportion of people living in poverty . . Then invnorm (.98, 0, 1)=2.05 . Round up to the next whole number. 2627 people
20) Here sigma is the population standard deviation of commuting distances for the Los Angeles residents. You first need to compute the standard deviation S using TI: stat>calc>edit to store the values, then stat>calc>one var stats: S=8.1 Next find . To compute . Df=8-1=7. Go to the Chi-Square table, . Now apply the formula to obtain , or
21) x is the number of American workers who think that they will be working when they are 65 or older. First find the mean and standard deviation using the binomial formulas: . Then apply normal correction a) . Normalcdf(-2.33, -0.78, 0, 1) b) . Normalcdf(.5, 100, 0, 1)
22)
a) x=# of questions answered correctly
n=5, p=1/4=0.15, q=1-0.25=0.75, x=4 or 5.
b) x=# of Latinos with diabetes
n=10, x=3 or higher, p=0.2, q=0.8
Use the complement, then subtract from 1: for the complement, use x=0, 1, or 2
23) x is the number of people who are in favor of junk mail ban. This is a binomial random variable. (Use the binomial formula)
24) The probability of drawing two hearts is . This is the probability of winning. Therefore, the probability of loosing is
X / P(x) / Xp(x)16 / 1/17 / 16/17
-4 / 16/17 / -64/17
(you may use decimal numbness for probability)
25) expected value = 1.5 variance. = 3-2.25=0.75, std dev =square root of 0.75
X / 0 / 1 / 2 / 3P(x) / 1/8 / 3/8 / 3/8 / 1/8
Xp(x) / 0 / .375 / .75 / .375 / 1.5
/ 0 / 1 / 4 / 9
/ 0 / 0.375 / 1.5 / 1.125 / 3
26) a) bell-shaped b) 0, 1 c) normal d) true e) true f) sampling, non-sampling g) 2.5
27)
is the population standard deviation of adult single-day ski lift tickets prices.
First find the sample standard deviation by: Stat/edit enter the values in L1. Then Stat/calc/one var stats/enter: s=10.39 Next go the chi-square table: , df=10-1=9.
28)
First apply the continuity correction: use the right end point 20.5. n=300, p=0.05, q=1-0.05=0.95.
Thus
(Use Normalcdf)
29)First observe that the net gain upon death is 20000-4000=16000 and the probability of death occurring during the year is 1-0.9=0.1
X / P(x) / Xp(x)-4000 / 0.9 / -3600
16000 / 0.1 / 1600
30) Use binomial It is easier to use the complement: P(at least 2)=1-(p(0)+p(1))=
31) First observe that the net wining is $12-$3=$9. Since there arfe 6 faces, the probability of winning is 1/6.
X / P(x)9 / 1/6=0.17 / 1.53
-3 / 5/6-0.83 / -2.49
32)X is the number of students enrolled in remedial math classes.
Use the binomial dist. to approximate the normal dist.
Use
33) Here you are asked to find x. First find the z-score before finding x: 1-0.13=0.87. Invnorm(0.87, 0, 1)=1.13. Next find x:
34) Use the binomial dist to approximate the normal dist.
X is the number of students who work full-time.
: using continuity correction, . Now . Thus
35) X=the time to complete an exam.
36a) true: the law of large numbers says approaches the population mean as n increases
b) false: the trials are independent,
c) false p is the parameter, is a statistic.
37) the population mean waiting time to see the professor.
invt(0.005. 7)= 3.499
First store the values in L1: Stat<Edit
Stat>Calc<One Var Stats
38) is the population mean speed of all cars entering the intersection of Grand and Temple.
First compute and using TI: stat>calc>edit to store the values, then stat>calc>one var stats:
Next compute : . Df=n-1=9. InvT(0.975, 9)=2.262
Thus a 90% confidence interval is
39)X= time spend studying per week
Use inverse normal since you are asked to find x. Get the area on the left side by 1-0.04=0.96. The z-score is invnorm(0.96, 0, 1)-=1.75. Now find x using
X=6.4+2(1.75)=9.9 hours.
40) This is a binomial problem
P(at least 2) means x=2 or higher. Thus it is easier to compute the complement (x=0 or x=1) and subtract the number from 1.
41)P= the population proportion of Mt. SAC students who are Baptists. First : . Invnorm(0.99, 0, 1)=2.33