# Probability Questions and Answers

Statistics

ProbabilitySuppose 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

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ProbabilityFor 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

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ProbabilityComplete 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

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ProbabilityConstruct 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

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ProbabilityThe 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

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ProbabilityStep 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

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ProbabilityYou 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

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Probabilitya 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

Statistics

ProbabilitySuppose 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

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ProbabilityMidsize 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

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ProbabilityA 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

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ProbabilityGiven 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

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ProbabilityGiven 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

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ProbabilityFind 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

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ProbabilityWhich 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

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Probability0 2 0 3 0 1 0 35 0 05 2 6 7 10 13 Find the expected value of the above random variable

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ProbabilityThe 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

Statistics

Probability69 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

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ProbabilityComplete 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

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ProbabilityThe 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

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Probability4 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

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ProbabilityStep 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

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ProbabilityThe 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

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ProbabilityA 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

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ProbabilityFind 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

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Probability21 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

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ProbabilityThe 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

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ProbabilityConsider 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

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ProbabilityThe 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

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Probability20 points A basketball player has a 62 chance of making each free throw What is the probability that the player makes exactly four out of six free throws 32 1 86 66

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ProbabilityA class has 20 students What is the probability that at least 5 students were born on a weekend 45 82 72

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Probability5 The diameter of chocolate chip cookies from a local bakery are normally distributed with an average diameter of 4 inches and a variance 2 25 Would a chocolate chip cookie from this local baker with a diameter of 8 inches be very unusual for this bakery Why or why not

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ProbabilityThree years ago the mean price of an existing single family home was 243 718 A real estate broker believes tha existing home prices in her neighborhood are higher a State the null and alternative hypotheses in words b State the null and alternative hypotheses symbolically c Explain what it would mean to make a Type I error d Explain what it would mean to make a Type II error a State the null hypothesis in words Choose the correct answer below OA The mean price of a single family home in the broker s neighborhood is greater than 243 718 OB The mean price of a single family home in the broker s neighborhood is different from 243 718 OC The mean price of a single family home in any neighborhood is 243 718 D The mean price of a single family home in the broker s neighborhood is 243 718 State the alternative hypothesis in words Choose the correct answer below OA The mean price of a single family home in the broker s neighborhood is different from 243 718 B The mean price of a single family home in any neighborhood is 243 718 C The mean price of a single family home in the broker s neighborhood is greater than 243 718 D The mean price of a single family home in the broker s neighborhood is 243 718 b State the hypotheses symbolically Ho H

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Probabilityo MY NOTES According to a 2017 Wired magazine article 40 of emails that are received are tracked using software that can tell the email sender when where and on what type of device the email w Wired magazine website Suppose we randomly select 40 received emails a What is the expected number of these emails that are tracked b What are the variance and standard deviation for the number of these emails that are tracked Round your answers to three decimal places Var x Need Help ASK YOUR TEACHER Read It PRACTICE A

Statistics

ProbabilityAre all batteries created equal An experiment was carried out to determine this Two brand name battery companies and a generic company were compared D batteries were put into flashlights and continuously left on The data provided is the number hours the flashlight remained lit Is this sufficient evidence to determine if the mean times for the different companies are different Use a 0 05 significance level What are the hypothesis statements for this situation H 2 G HA at least one mean is different Ho P1 P2 PG HA at least one proportion is different Ho Battery type and time are independent HA Battery type and time are dependent

Statistics

ProbabilityStep 9 is not sufficient evidence to conclude that the population proportion is less than 0 85 p 0 85 c p 0 80 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 80 Use these values to find the test statistic rounding the result to two decimal places Z P Po Po 1 PO n 0 80 0 85 0 85 10 85 160 Enter a number Submit Skip you cannot come back

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ProbabilityMatch each scatterplot shown below with one of the four specified correlations 9 O 8 9 Q OQ a 0 29 b 0 90 c 0 97 d 0 25

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Probabilityeach scatterplot shown below with one of the four specified correlations 00 O sg 8 5 0 80 oo 00 8 oo 080 80 80 08 0 Q Q a 0 99 b 0 70 c 0 13 d 0 16

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ProbabilityIf there is strong correlation between two variables it is definitely true that one is causing the other O False True

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ProbabilityHere are the scatter plots for two sets of bivariate data with the same response variable The first compares the variables x y The second compares the variables w y O 8 O Questi O O O O 80 Q Which explanatory variable has a stronger relationship with the response variable y O The first variable x has a stronger relationship with the response variable y O The second variable w has a stronger relationship with the response variable y

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ProbabilityA correlation of r 85 is found between weekly sales of firewood and cough drops over a 1 year period Which of the following is true O a There is a pretty strong positive linear relationship between sales of firewood and cough drops O b Fire must be the cause of coughing c Temperature is a possible lurking variable that is behind this relationship O d Both a and c are true O e None of the above

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ProbabilityNegative linear relationships are represented by values of the correlation r that are O greater than zero O less than zero O zero

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ProbabilityYou may need to use the appropriate appendix table or technology to answer this question Sixty eight percent of online courses taught at community colleges nationwide were taught by full time faculty To test if 68 also represents California s percent for full time faculty teaching the online classes Long Beach City College LBCC in California was randomly selected for comparison In the same year 34 of the 44 online courses LBCC offered were taught by full time faculty Conduct a hypothesis test at the 5 level to determine if 68 represents California State the null and alternative hypotheses Enter for as needed Ho 0 68 Ha 0 68 X X What is the test statistic Round your answer to two decimal places What is are the critical value s If using the z distribution round your answer s to two decimal places and if using the t distribution round your answer s to three decimal places Enter NONE for any unused answer blanks lower tail upper tail What is the decision of the test and what conclusions can be drawn O At the 5 level of significance we would reject Ho and conclude that there is sufficient evidence to conclude that the percent of online courses taught at community colleges in the state is not equal to 68 O At the 5 level of significance we would fail to reject Ho and conclude that there is sufficient evidence to conclude that the percent of online courses taught at community colleges in the state is not equal to 68 O At the 5 level of significance we would fail to reject Ho and conclude that there is not sufficient evidence to conclude that the percent of online courses taught at community colleges in the state is not equal to 68 At the 5 level of significance we would reject Ho and conclude that there is not sufficient evidence to conclude that the percent of online courses taught at community colleges in the state is not equal to 68

Statistics

ProbabilityRecall 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 82 Use these values to find the test statistic rounding the result to two decimal places Z P Po Po 1 PO n 0 82 0 85 1 0 85 160 Submit Skip you cannot come back

Statistics

ProbabilityThe value for po is the hypothesized population proportion we are testing for Here we have po n 160 0 85 Z P Po Po 1 Po n 160 Step 2 Now that all of the values have been identified the test statistic can be calculated Use p 0 78 p 0 85 and n 160 to find the value of the z test statistic rounding the result to two decimal places 0 78 0 85 0 85 1 0 85 160 There was a sample of 160 items given so we have

Statistics

ProbabilityYou may need to use the appropriate appendix table or technology to answer this question At 95 confidence how large a sample should be taken to obtain a margin of error of 0 08 for the estimation of a population proportion Assume that past data are not available for devele planning value for p Round your answer up to the nearest whole number

Statistics

Probabilitya p 0 78 Before a p value can be found the test statistic must be calculated and the type of test must be determined The following hypotheses were given for a sample of 160 items and a sample proportion of p 0 78 Since this test is for a population proportion the z test statistic will be found This calculation is given below where p is the sample proportion Po is the hypothesized population proportion and n is the sample size Ho P 0 85 Ha P 0 85 Submit Z The value for po is the hypothesized population proportion we are testing for Here we have po Need Help P Po Po 1 PO n Skip you cannot come back Read It There was a sample of 160 items given so we have n

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ProbabilityThe null hypothesis Ho A states that the difference between sample means is statistically significant B states that the difference between sample means is due to the independent variable D cannot be tested directly states that the difference between sample means is due to chance I

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Probability8 pts We want to estimate the proportion of University of Montana Missoula College students who support a new bill regarding minimum wage We collect a simple random sample from 200 students and 109 say they support the new bill Construct a 95 confidence interval estimating the proportion of UM MC students who support the bill a What is our sample statistic p