Probability Questions and Answers

The national average SAT score is roughly 1500 We would like to see if the high school seniors who apply to Duke have higher than average SAT scores We randomly sample 100 applicants files and record their SAT scores Which of the following is the correct set of hypotheses for this research question Hint think about how many samples we have data from O Ho x 1500 HA 1500 O Ho H 1500 HA H 1500 O Ho P 1500 HA P 1500 O Ho H 1500 HA H 1500 Ho H Duke H National HA H Duke National

More than 18 of people support legalizing gay marriage Express the null and alternative hypotheses in symbolic form for this claim enter as a percentage Ho P H p Use the following codes to enter the following symbols enter

The mean weight of male nurses working at a local hospital is more than 174 lbs Express the null and alternative hypotheses in symbolic form for this claim Ho p H p Use the following codes to enter the following symbols enter

3x 16 x 2 X

Solve 9 x 7 4 10 for x

Use the quadratic formula to solve the equation Be sure to simplify your answers Enter both answers separated by a comma 4x 64 0

Solve the proportion 29 X 26 42 Round to two decimal places if necessary

Solve the proportion 38 46 48 8

Solve the equation 6x 10 3 algebraically If your answer is a fraction write it in reduced fraction form Do NOT convert the answer to a decimal X

1 5 x 4 8x 10

6 A box contains 4 red and 3 green ornaments TWO ornaments are randomly selected without replacement Find the indicated probabilities A What is the probability both ornaments are red B What is the probability that the 1st one is red and the 2nd one is green C What is the probability both ornaments are the same color

E 5 A store offers a special Christmas purchase in which for a fixed cost you can choose from 1 of two pants 1 of four shirts and 1 of five ties How many different outfits are available

In the class we have discussed about how to derive the probability of type ll for the lower tailed test Please follow the similar procedure and derive the probability of type Il error B u for the upper tailed test and two tailed test Alternative Hypothesis 09 Hai 0 Hai Ho Ha H0 Type II Error Probability for a Level a Test 20 10 za 10 1 20 2 140 1 9 30 2 160 1 2a

A company has a policy of retiring company cars this policy looks at number of miles driven purpose of trips style of car and other features The distribution of the number of months in service for the fleet of cars is bell shaped and has a mean of 47 months and a standard deviation of 11 months Using the empirical rule as presented in the book what is the approximate percentage of cars that remain in service between 25 and 36 months Do not enter the percent symbol ans

Chapter 3 3 4 The probability of an event and the probability of its complement always sum to a 1 d between 0 and 1 b o c 1 The area under the curve to the right of z 0 is a 0 6816 b 1 0000 c 0 5000 d o

Construct a Binomial probability distrubtion table using n 5 and p 0 23 Next record the mean an standard deviation of the distribution below Where necessary round all numbers to four decimal places O 2 3 ST 5 1 15 x 0 2648 0 3794 0 2424 0 0886 0 0208 0 0021 P x X X X X X X

Can the following table be a probability distribution P x 1 0 3256 0 1755 0 1402 4 0 234 0 1247 8 23WN 5

A fair coin is flipped 14 times Find the probability that more than 6 of the flips turn up tails 0 604736 0 453879 0 633746 0 545104 0 662034

In a recent survey 35 indicated chocolate was their favorite flavor of ice cream Suppose we select a sample of nine people and ask them to name their favorite flavor of ice cream a How many of those in the sample would you expect to name chocolate Round answer to 2 decimal places Expected number of people b What is the probability exactly six of those in the sample name chocolate Round answer to 4 decimal places Probability c What is the probability that six or more name chocolate Round answer to 4 decimal places Probability

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of 4 emails per hour Assume the arrival of these emails is approximated by the Poisson distribution a What is the probability Linda Lahey company president received exactly 1 email between 4 P M and 5 P M yesterday Round answer to 4 decimal places Probability b What is the probability she received 5 or more emails during the same period Round answer to 4 decimal places Probability c What is the probability she did not receive any emails during the period Round answer to 4 decimal places Probability

Suppose you wish to find out the answer to the age old question Do people prefer Cola 1 or Cola 2 You conduct a blind taste test in which individuals are randomly asked to drink one of the colas first followed by the other cola and then asked to disclose which drink they prefer Results of your taste test indicate that 47 of 100 individuals prefer Cola 2 Complete parts a through c a Conduct a hypothesis test preferably using technology Ho p Po versus H p po for po 0 36 0 37 0 38 0 56 0 57 0 58 at the x 0 05 level of significance For which values of po do you not reject the null hypothesis What do each of the values of po represent Do not reject the null hypothesis for the values of po between and inclusively Type integers or decimals Do not round

For every football game there is a team that is expected to win by a certain number of points In betting parlance thi is called the spread If point spreads are accurate we would expect about half of all games played to result in the favored team winning beating the spread and about half of all games to result in the team favored to not beat the spread The accompanying data represent the results of 45 randomly selected games where a 0 indicates the favor team did not beat the spread and a 1 indicates the favored team beat the spread Do the data suggest that sport books establish accurate spreads Click the icon to view the spread results Because npo 1 Po Round to one decimal place as needed Spread Results 110 00 1 1 000 0 0 0 10 the sample size is 1 0 0 0 1 0 1 1 0 1 1 5 of the population size and the sample the requirements for testing the hypothesis satisfied 00 1 1 1 0 1 1 10 X

Complete parts a through c below of variables were measured for all tornadoes in the United States in 2017 Ho H a Since 1950 the proportion of tornadoes that have been FO is 0 465 Test whether a different proportion of FO tornadoes occur in Texas than nationally Note Treat the tornadoes that struck in Texas as a simple random sampl all tornadoes that struck Texas since 1950 using a 0 05 level of significance The summarized data show that 82 o 168 tornadoes in Texas were FO during 2017 State the null and alternative hypotheses 0 of 1 Type integers or decimals Do not round Save

Construct a Binomial probability distrubtion table using n 5 and p 0 23 Next record the mean and standard deviation of the distribution below Where necessary round all numbers to four decimal places P x l b 11 x

1 Solve the triangle with the given angles and sides 6 points B 52 C 28 a 43c

The time required to assemble a part of a machine follows an exponential probability distribution with a mean of 16 minutes Round your answers to four decimal places a What is the probability that the part can be assembled in 9 minutes or less b What is the probability that the part can be assembled between 4 5 and 9 minutes Need Help Read It

Step 2 This is a binomial experiment with n 18 trials Let a success be that a student withdraws from the course It is given that 15 of students withdraw Recall the method used to find the probability of an event given a percentage probability Thus the probability p a student withdraws from the course is as follows given percentage probability 100 p 0 15 given percentage 100 0 4797 100 X X For 2 or fewer students to withdraw from the course means that exactly Select students withdraw Therefore x will be 0 and 1 probability that 2 or fewer students withdraw from the course will be Select Submit Skip you cannot come back X The probability of each x value is f x so the

You may need to use the appropriate appendix table or technology to answer this question Consider a binomial experiment with n 10 and p 0 10 a Compute f 0 Round your answer to four decimal places f 0 0 3487 b Compute f 2 Round your answer to four decimal places f 2 0 9995 X c Compute P x 2 Round your answer to four decimal places P x 2 0 1937 d Compute P x 1 Round your answer to four decimal places P x 1 e Compute E x E x f Compute Var x and o Round your answer for o to two decimal places Var x

a Compute the probability that none of the appeals will be successful Since there are only two outcomes this can be thought of as a binomial experiment Recall the binomial probability function below where x is the number of success p is the probability of a success on one trial and there are n trials probability A sample of 10 first round appeals have been received so we have n Let p be the probability that a first round appeal was successful It is given that 35 of first round appeals were successful This percentage must be converted to a probability as follows P Submit f x p 1 p x n x n x percentage 100 35 100 p n x where Of interest is the probability that none of the appeals will be successful Letting x be the number of successful appeals the probability that x Skip you cannot come back is to be found

Suppose you model a game of chance with a discrete probability distribution Let X be the net amount o money won or lost by the player Let P X be the probability of the corresponding outcome The three events are as follows There is a 23 chance the player wins 3 dollars There is a 47 chance the player breaks even There is a 30 chance the player loses 6 dollars Complete the table below to model the scenario X P X

Midsize Large and SUV with measured amounts of left leg femur force kN resul in the following Minitab display Using a 0 05 significance level test the claim that th four vehicle size categories have the same mean force on the femur of the left leg Does size of the car appear to have an effect on the force on the left femur in crash tests Analysis of Variance Source Size Error Total DF 3 52 55 Ho Adj SS 0 6381 20 4828 21 1209 Determine the null hypothesis H1 H2H3 H4 Adj MS 0 2127 0 3939 All of the means are different H H H H F Value 0 54 At least one of the means is different from the others P Value 0 657

A data set includes the counts of chocolate chips from three different types of Chips Ahoy cookies The accompanying StatCrunch display shows results from analysis c variance used with those three types of cookies Use a 0 05 significance level to tes the claim that the three different types of cookies have the same mean number of chocolate chips ANOVA table Source DF Columns Error Total SS F Stat P value MS 886 9875 443 49375 57 7981 0 0001 Ho 2 77 590 83333 7 6731601 79 1477 8208 Determine the null hypothesis

Given the following Linear Regression model Pounds of Trash Per Household 1 5 x Person 1 5 How many pounds of trash are created by a household of 5 Edit View Insert Format Tools Table

Given the following Linear Regression model Pounds of Trash Per Household 1 5 x Person 1 5 What is the slope Edit View Insert Format Tools Table

Find the standard deviation for the random variable X 5 3 7 9 14 P X x 0 23 0 23 0 24 0 15 0 15 X 6 165887121899 5 1248185333727 6 2192968252046 6 0422483398152

Which of the following is NOT true Choose the correct answer below CERED O A When testing for equality of three population means do not use multiple hypothesis tests with two samples at a time OB As the number of tests of significance increase the risk of finding a difference by chance alone increases OC If the decision from ANOVA is to reject the equality of the three population means the particular mean that differs from the others is kno OD The F test statistic is very sensitive to sample means even though it is obtained through two different estimates of the common population variar

0 2 0 3 0 1 0 35 0 05 2 6 7 10 13 Find the expected value of the above random variable

The Insure com website reports that the mean annual premium for automobile insurance in the United States was 1 503 in March 2014 Being from Pennsylvania at that time you believed automobile insurance was cheaper there and decided to develop statistical support for your opinion A sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of 1 430 with a standard deviation of s 160 a Develop a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium O Hoi R 1 503 H 1 503 Ho 1 503 Ha 21 503 Ho 2 1 503 Ha 1 503 Hot S 1 503 H 1 503 Ho 1 503 H H 1 503 b What is a point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean Use the mean annual premium in Pennsylvania minus the national mean c At a 0 05 test for a significant difference Find the value of the test statistic Round your answer to three decimal places Find the p value Round your answer to four decimal places p value State your conclusion O Reject Ho We cannot conclude that the population mean automobile premium in Pennsylvania is lower than the national mean O Do not reject Ho We cannot conclude that the population mean automobile premium in Pennsylvania is lower than the national mean O Reject Ho We can conclude that the population mean automobile premium in Pennsylvania is lower than the national mean Do not reject Ho We can conclude that the population mean automobile premium in Pennsylvania is lower than the national mean

69 A statistics instructor believes that fewer than 20 percent of Evergreen Valley College EVC students attended the opening midnight showing of the latest Harry Potter movie She surveys 84 of her students and finds that 11 of them attended the midnight showing The Type I error is to conclude that the percent of EVC students who attended is a at least 20 percent when in fact it is less than 20 percent b 20 percent when in fact it is 20 percent c less than 20 percent when in fact it is at least 20 percent d less than 20 percent when in fact it is less than 20 percent

Complete the truth table for the given statement by filling in the required columns qvp pq TT TF FT F F q Complete the truth table qvp

The circle graph shows the most important problems for the 19 175 341 high school teenagers in a country Without using a calculator estimate the number of high school teenagers for whom doing well in school is the most important problem Most Important Problems for High School Teenagers Doing Well in School 12 Social Pressures and Fitting in 21 OA 1 900 000 OB 5 700 000 OC 3 800 000 O D 7 600 000 Drugs 22 Others 29 Getting along with Parents 2 Getting into College 4 Sexual Issues 4 Crime and Violence in School 68 Doing well in school is the most important problem for approximately how many of the high school teenagers

4 Create a sample tree showing conditional probabilities for drawing 4 balls randomly out of a sock with 3 green and 7 yellow balls without replacement S JOLAS PASS

Step 11 If the found p value is less than or equal to a given level of a then the null hypothesis will be rejected in favor of the alternative The p value was found to be 0 0384 and we are to compare the p value to a 0 05 Since the p value is less than or equal to is sufficient evidence to conclude that the population proportion is less than 0 85 p 0 85 Step 12 d p 0 89 Z Recall the previously determined values po 0 85 and n 160 Now we are to find the test statistic for a new sample proportion p 0 89 Use these values to find the test statistic rounding the result to two decimal places P Po Po 1 Po n 0 89 less than or equal to 0 05 the null hypothesis Hor will 0 85 10 85 160 will be rejected in favor of the alternative Therefore there is

The bar graph shows that life expectancy the number of years newborns are expected to live in a particular region has increased dramatically since ancient times Find the percent increase in average life expectancy of the region from the Stone Age to 2016 Round to the nearest integer as needed Average Life Expectancy years 80 70 60 50 40 30 20 10 0 27 29 32 34 52 63 72 Stone Ancient Middle 1900 1950 1980 2016 Greece Ages Age Time Period Year Q 17

A single die is rolled twice The 36 equally likely outcomes are shown to the right Find the probability of getting two numbers whose sum is 2 First Roll Second Roll 1 1 1 2 1 3 1 4 1 5 1 6 2 1 2 2 2 3 2 4 2 5 2 6 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 5 3 5 4 5 5 5 6 6 1 6 2 6 3 6 4 6 5 6 6 The probability of getting two numbers whose sum is 2 is Type an integer or a simplified fraction

Find a the mean b the deviation from the mean for each data item and c the sum of the deviations in part b for the following group of data items 57 67 62 57 72 a The mean is Simplify your answer b Complete the following table by entering the deviations for each data value 57 67 62 Simplify your answers c The sum of the deviations is Simplify your answer 57 72

21 A day of the week is to be chosen at random from the 7 day week Let event A be that a day from the 5 day work week was chosen Let event B be that the day of the week chosen starts with the letters S or T Find P A and P B Are events A and B mutually exclusive Why

The time required to assemble a single Play Station 5 is approximately normally distributed with 15 44 minutes and o 1 2 minutes A random sample of 19 Play Station 5 assembly times is selected Find the minimum assembly time of the upper 10 of average assembly times Round to at least 2 decimal places if needed

Consider the density curve plotted below PDF X 0 275 0 25 0 225 Q3 0 2 IQR 0 175 0 15 0 125 0 1 0 075 0 05 0 025 Find P X 18 Find P X 19 15 Calculate the following Q1 median 16 17 Density Curve 18 X 9 20 a

The following table is for all pets in the village of MakeBelieve Age 7 years Age 7 years Totals Dogs Cats Totals In Make Believe the probability the age of a pet more than 7 years old is 0 4 for a dog and 0 6 for a cat Complete the table Compute the probability that the pet is a cat given it is not more than 7 years old