Statistics Wk 4

On average, Roger has noticed that 9 runners pass by his house each day. What is the probability that exactly 5 runners will pass his house in a day? (Answers are rounded to the thousandths digit.)

0.061 The time interval of interest is a day. The average number is 9 per day, so if X is the number of runners that Roger sees in a day, then X∼Po(λ) where λ=9. We want to know the probability of 5 runners in a day, which means x=5. According to the formula, we find P(X=x)=λxe−λx!=95e−95!≈0.061

According to a 2011 Pew Research poll, cell owners make or receive an average of 12 phone calls each day. Let X = the number of phone calls that smartphone users send or receive per day. The random variable X has a Poisson distribution: X ~ P(12). What is the probability that a smartphone user makes or receives exactly 20 phone calls in 2 days? (Round to the thousandths place.)

0.062 To find the probability of a Poisson Distribution, use the formula: f(x)=e−μ⋅μxx! ...where μ is the average, and x is the number of successes the question is asking about. In this question, the value of x is 20 because that is the number of phone calls (or successes) that the question is asking about. However, the interval is changed from 1 day to 2 days, so μ should be doubled. The average number of phone calls per day is 12, and 12⋅2=24. So the new value of μ is 24. Plugging these values into the equation, you get: f(x)=e−24⋅242020! ...which simplifies to 0.062, rounded to the thousandths digit.

In a recent basketball season, Jenny sunk a three-point shot once in every 3.5 attempts, on average. Assume that this probability did not change going into the next season. What is the probability that Jenny sinks her first three-point shot on her third attempt of the season? Use Excel to find the probability. Round your answer to three decimal places.

0.146​ Use Excel to find the probability. Note that this is not a cumulative probability. Let a success be Jenny sinking a three-point shot. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of three trials, there are 2 failures before the first success. Thus, enter 2, 1, and 1/3.5, in that order. In the entry for a cumulative probability, enter 0 since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.145773, which is 0.146 rounded to three decimal places.

A certain cold remedy has an 88% rate of success of reducing symptoms within 24 hours. Find the probability that in a random sample of 45 people who took the remedy, 40 of them had a reduction of symptoms within a day. Round your answer to three decimal places.

0.183​ Note that this is a binomial probability. In this case, we want to find the probability of 40 successes, where a success is a reduction of symptoms. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 40, 45, and 0.88, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.18289, which is 0.183 rounded to three decimal places.

According to a recent poll, smart phone users receive an average of 4 phone calls each day. Let X = the number of phone calls that smartphone users receive per day. The random variable X has a Poisson distribution: X ~ P(4). What is the probability that a smartphone users receives exactly 3 phone calls per day? (Rounded to the thousandths place.)

0.195 To find the probability of a Poisson Distribution, use the formula: f(x)=e−μ⋅μxx! ...where μ is the average, and x is the number of successes the question is asking about. In this question, μ is 4, the average number of phone calls smartphone users receive in a day. The value of x is 3 because that is the number of phone calls (or successes) that the question is asking about. Plugging these values into the equation, you get: f(x)=e−4⋅433! ...which simplifies to 0.195, rounded to the thousandths digit.

65% of the people in Missouri pass the driver's test on the first attempt. A group of 7 people took the test. What is the probability that at most 3 in the group pass their driver's tests in their first attempt? Round your answer to three decimal places. Remember: 65% = 0.65.

0.200 The parameters of this binomial experiment are: n = 7 trials p = 0.65 x = at most 3 successes This means that x takes on more than one value. At most 3 means that x can be 0,1,2, or 3 successes. Therefore, the binomial probability is: P(X≤3)=P(X=0)+P(X=1)+P(X=2)+P(X=3) =7C0(0.65)0(1−0.65)7−0+7C1(0.65)1(1−0.65)7−1+7C2(0.65)2(1−0.65)7−2+7C3(0.65)3(1−0.65)7−3 =7!0!(7−0)!(0.65)0(1−0.65)7+7!1!(7−1)!(0.65)1(1−0.65)6+7!2!(7−2)!(0.65)2(1−0.65)5+7!3!(7−3)!(0.65)3(1−0.65)4 ≈0.001+0.008+0.047+0.144 =0.200 So, the probability that at least 3 people pass their driver's test on their first attempt is approximately 0.200 or 20.0%.

A baseball park ran a "$1 Hot Dog Night" promotion. The average number of hot dogs purchased that night was 3.3 hot dogs per fan. Find the probability that a randomly selected fan who attended $1 Hot Dog Night purchased 5 or more hot dogs. Round your answer to 3 decimal places.

0.237​ Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 5 or more occurrences for a fan. This is the complement of the probability of 0 to 4 occurrences for a fan, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 4 and 3.3, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.76259, which is 0.763 rounded to three decimal places. To find the probability of the fan buying 5 or more hot dogs, subtract this probability from 1. The probability is 1−0.763=0.237.

The National Coffee Association claimed that in 2010, Americans drank an average of 3 cups of coffee per day. Let X = the number of cups of coffee Joe drinks in a day. The random variable X has a Poisson distribution: X∼Po(3) . What is the probability that he drinks 2 or fewer cups of coffee in a day? (Answers are rounded to the thousandths digit.)

0.423 The average is 3, so the parameter is λ=3. Remember that the formula is P(X=x)=λxe−λx! The probability of drinking 2 or fewer cups is P(X≤2)=P(X=0)+P(X=1)+P(X=2)=30e−30!+31e−31!+32e−32!≈0.050+0.149+0.224≈0.423

Give the numerical value of the parameter p in the following binomial distribution scenario. A softball pitcher has a 0.721 probability of throwing a strike for each pitch and a 0.279 probability of throwing a ball. If the softball pitcher throws 19 pitches, we want to know the probability that more than 15 of them are strikes. Consider strikes as successes in the binomial distribution. Do not include p= in your answer.

0.721​ The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a strike, so p=0.721.

At a certain fast food restaurant, 77.5% of the customers order items from the value menu. If 14 customers are randomly selected, what is the probability that at least 9 customers ordered an item from the value menu? Use Excel to find the probability. Round your answer to three decimal places.

0.927​ Note that this is a binomial probability. In this case, we want to find the probability of 9 to 14 successes, inclusive, where a success is a customer ordering from the value menu. This probability is the complement of the probability of 0 to 8 successes, inclusive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 8, 14, and 0.775, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.072766, which is 0.073 rounded to three decimal places. To find the probability of 9 to 14 customers ordering from the value menu, subtract the previous probability from 1. The probability is 1−0.073=0.927

For the below problem, which values would you fill in the blanks of the function B(x;n,p)? The probability of saving a penalty kick from the opposing team is 0.617 for a soccer goalie. If 7 penalty kicks are shot at the goal, what is the probability that the goalie will save 5 of them?

B(5;7,0.617) The parameters of a binomial distribution are: n = the number of trials x = the number of successes in the whole experiment p = the probability of a success The parameters should be in the order of x, n, p in the binomial function B(x;n,p). So, in this case, you should input B(5;7,0.617).

An insurance company is assessing the risk of selling flood insurance. Under the company's policies, if a customer lives in an area where they have a greater than 5% chance of experiencing 4 or more floods in 10 years, the customer must pay a higher premium. Drew lives in an area where the average number of floods per 10 years is 1.6 floods. Will Drew have to pay the higher premium for flood insurance? Use Excel to find the probability.

Drew will have to pay the higher premium because the probability is greater than 0.05. Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 4 occurrences or more for 10 years. This is the complement of the probability of 3 occurrences or fewer. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 3 and then 1.6, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.921187, which is 0.921 rounded to three decimal places. To find the probability of 4 or more floods, subtract this probability from 1. The probability is 1−0.921=0.079. Since this probability is greater than 0.05, Drew will have to pay the higher premium.

Jackie is practicing free throws after basketball practice. She makes a free throw shot with probability 0.7. She takes 20 shots. We say that making a shot is a success. What are p, q, and n in this context?

p=0.7, q=0.3, n=20 Remember that p is the probability of success, which is the probability of making a shot, 0.7. The probability of failure is q=1−p=0.3. n is the number of repetitions, which is 20.

Burt, a football quarterback, has a pass completion percentage of 55.2%. If the probability that Burt will need 8 or more pass attempts to complete his first pass in a game is at least 0.01, then Burt will be benched and replaced by a backup. Will Burt be benched? Use Excel to find the probability, rounding to three decimal places.

probability that Burt will need 8 or more pass attempts to complete his first pass =P(not make a pass in first 7 attempts) =(1-0.552)7 =0.004 as this is less than 0.01 therefore Burt will not be benched********

Mean of a geometric distribution

tells you how many trials you should expect to go through until a success occurs The Mean of a geometric distribution may also be referred to as the Expected value of a geometric distribution

Compliment rule

the probability of an event equals 1 minus the probability of its complement (that the event will not happen)

To see if a spinner that is divided into 100 equal sections labeled 1 to 100 is fair, a researcher spins the spinner 1000 times and records the result. Let X represent the outcome. The table below shows the probability distribution of the data. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to two decimal places. x P(x) 1 0.011 2 0.011 3 0.011 4 0.01 5 0.008 6 0.011 7 0.011 8 0.01 9 0.01

{mean}=50.28 {, standard deviation}=mean=50.28, standard deviation=28.99​ The mean and the standard deviation of the probability distribution can be calculated using Excel. Step 1: Place the values of the random variable, x, in column A and the corresponding probabilities in column B. Then find the product of column A and column B. Step 2: Add the elements in the third column to find the mean, μ, rounded to two decimal places, which is 50.28. Step 3: To find the standard deviation, add column D to calculate the product of the square of column A and column B. Now add the elements in the fourth column to find ∑x2P(x), which is 3368.351. Step 4: Calculate the variance by subtracting the square of the mean from ∑x2P(x). The variance, rounded to three decimal places, is 840.574. Step 5: Find the square root of the variance to find the standard deviation. The standard deviation, σ, rounded to two decimal places, is 28.99.

Christine has a motion detector light which gets activated an average of 16 times every 2 hours during the night. In order to find the probability that the motion detector light will be activated more than 4 times in a 25 minute period during the night using the Poisson distribution, what is the average number of activations per 25 minutes? Round your answer to three decimal places.

λ=3.333​ The ratio of the average number of activations in 2 hours is equal to the ratio of the average number of activations in 25 minutes. 2 hours is equivalent to 120 minutes. Let λ represent the average number of activations in 25 minutes, then λ25=16120. Solving this equation for λ shows that λ=16(25)120=3.333.

Elizabeth gets an average of 10 emails during her 8 hour work day. In order to find the probability that Elizabeth will get more than 3 emails in a 3 hour portion of her work day using the Poisson distribution, what is the average number of emails received per 3 hours? Round your answer to two decimal places.

λ=3.75​ The ratio of the average number of emails in 8 hours is equal to the ratio of the average number of emails in 3 hours. Let λ represent the average number of emails in 3 hours, then λ3=108. Solving this equation for λ shows that λ=10(3)8=3.75.

According to a Gallup poll, 60% of American adults prefer saving over spending. Let X= the number of American adults out of a random sample of 50 who prefer saving to spending. What is the mean (μ) and standard deviation (σ) of X?

μ=30 and σ≈3.46 Remember that the mean μ is given by the formula μ=np. This should make sense because you can think of p as the fraction of the sample, on average, that will be a success. In this case p=0.6 because we think of a success as someone who prefers saving over spending. n is the size of the sample, 50. So μ=(50)(0.6)=30 Standard deviation is given by the formula σ=npq‾‾‾‾√. As above, n=50 and p=0.6. Remember that p+q=1, so solving for q we find that q=1−p=0.4. So σ=(50)(0.6)(0.4)‾‾‾‾‾‾‾‾‾‾‾‾‾√=12‾‾‾√≈3.46

Identify the parameters in the following situation. When Jackie is camping, she sees 6 shooting stars every hour, on average. What is the probability that she sees exactly 4 shooting stars in the next hour?

μ=6; x=4; time period is one hour We are asked for the probability of 4 shooting stars in one hour. Therefore, the time period of interest is 1 hour and the value of interest is x=4. The average value μ is given to be 6 shooting stars per hour. So the answer is μ=6; x=4; and the time period is 1 hour.

binomial experiments

Experiments in which only two outcomes are possible x successes in n tries

Z-score

a measure of how far a point is from the mean, measured in standard deviations, denoted by the letter z Z-score is also referred to as the Test statistic

Sample space

all possible outcomes or results for an experiment

Empirical rule

for a normal distribution, nearly all of the data will fall within 3 standard deviations of the mean, with 68% of the data falling within one standard deviation, 95% falling within two standard deviations, and 99.7% within three standard deviations The Empirical rule is also know as the 68-95-99.7 rule, the Standard deviation rule, or the Three sigma rule

Poisson experiment

is a random experiment in which the number of occurrences of a given event during a specified period of time is observed. The occurrences of the event are assumed to be random and independent one to another x successes during an interval

Standard normal table

tells you the probability of a z-score less than the value in the table, which gives you the area under the normal curve to the left of the value in the table A Standard normal table is commonly referred to as a Z-table, a Unit normal table, or a Normal probability table

Law of large numbers

the more trials you perform, the closer and closer the actual results get to the theoretical probabilities predicted for an experiment

Remember: When the intervals are different....

you must multiply the mean (μ) by the proportion, creating a new value for μ to be used in the Poisson formula

Expected value

the predicted outcome of an experiment after repeating over a long period of time, usually denoted by the Greek letter μ The Expected value is also known as the Expectation, "Long-term" average, or "Long-term" mean

Binomial

an experiment with a set of identical trials each with only two possible outcomes Binomial trial is also known as a Bernoulli trial

Discrete random variable

variables whose possible values are a list of distinct values

Discrete random variable:

variables whose possible values are a list of distinct values

How to Find the Probability of a Binomial Distribution

Use the formula: B(x;n,p)=nCx⋅px⋅(1−p)n−x Note: nCx=n!x!(n−x)!

Give the numerical value of the parameter n in the following binomial distribution scenario. The probability of buying a movie ticket with a popcorn coupon is 0.597 and without a popcorn coupon is 0.403. If you buy 18 movie tickets, we want to know the probability that no more than 13 of the tickets have popcorn coupons. Consider tickets with popcorn coupons as successes in the binomial distribution. Do not include n= in your answer.

18 The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, the total number of trials, or movie tickets, is n=18.

Suppose X∼N(9,1.5), and x=13.5. Find and interpret the z-score of the standardized normal random variable.

The z-score when x=13.5 is: _______ The mean is: ___________ This z-score tells you that x=13.5 is standard deviations to the right of the mean.

Jamie is practicing free throws before her next basketball game. The probability that she makes each shot is 0.6. If she takes 10 shots, what is the probability that she makes exactly 7 of them? Round your answer to three decimal places.

0.215 Here, the parameters are x=7, p=0.6, and n=10. Plugging this in to the formula for B(x;n,p), we find B(x;n,p)=nCx⋅px⋅(1−p)n−x=10C7⋅(0.6)7⋅(1−0.6)3=10!3!7!⋅(0.6)7⋅(0.4)3≈120⋅(0.028)⋅(0.064)≈0.215 So the probability of making exactly 7 shots is approximately 0.215.

In a recent baseball season, Bob hit a home run approximately once every 18.38 plate appearances. Assume that this probability did not change going into the next season. What is the probability that Bob hits his first home run before his 25th plate appearance of the season? Use Excel to find the probability. Round your answer to three decimal places.

0.739​ Use Excel to find the probability. Note that this is a cumulative probability. Let a success be hitting a home run. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most 24 trials, there are at most 23 failures before the first success. Thus, enter 23, 1, and 1/18.38, in that order. In the entry for a cumulative probability, enter 1 since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.738843, which is 0.739 rounded to three decimal places.

Mark receives 10 calls in 3 days on average. How many calls does he receive in 5 days on average?

16.7 calls If μ is the unknown average number of calls over 5 days, then we can set up the proportion: 10 calls3 days=μ5 days Cross multiplying, we find that 3μ=50, so solving for μ, we find that μ=503≈16.7 calls

In baseball, the statistic Walks plus Hits per Inning Pitched (WHIP) measures the average number of hits and walks allowed by a pitcher per inning. In a recent season, Burt recorded a WHIP of 1.315. Find the probability that, in a randomly selected inning, Burt allowed a total of 3 or more walks and hits. Use Excel to find the probability. Round your answer to three decimal places.

Note that this is a cumulative Poisson probability. In this case, we want to find the probability of 3 or more occurrences. This is the complement of the probability of 0 to 2 occurrences for one inning, inclusive. To determine the cumulative probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 2 and 1.315, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.853644, which is 0.854 rounded to three decimal places. To find the probability of 3 or more walks plus hits, subtract this probability from 1. The probability is 1−0.854=0.146.

You are told that a data set has a median of 64 and a mean of 52. Which of the following is a logical conclusion?

The data are skewed to the left Because the mean, 52, is less than the median, 64, we expect that there are some very small values which are bringing the mean down. In other words, the data are skewed to the left.

Geometric distribution

consists of one or more Bernoulli trials with each trail resulting in a failure except the last one, which is a success; in other words, you keep trying until you get a success

Probability distribution

contains the probability of each possible outcome for a discrete random variable and always adds up to 1.0

Discrete data

data that you can count

According to a recent poll, smart phone users receive an average of 4 phone calls each day. Let X = the number of phone calls that smartphone users receive per day. The random variable X has a Poisson distribution: X ~ P(4). What is the probability that a smartphone user receives at most 2 phone calls per day? (Round to the thousandths place.)

0.238 To find the probability of a Poisson Distribution, use the formula: f(x)=e−μ⋅μxx! ...where μ is the average, and x is the number of successes the question is asking about. In this question, μ is 4, the average number of phone calls smartphone users receive in a day. The value of x is 2 because that is the number of phone calls (or successes) that the question is asking about. But since the question asks for "at most 2," you must find the probility of P(X=0)+P(X=1)+P(X=2), all of the possible values of 2 or less. Plugging these values into the equation, you get: f(x)f(x)=P(X=0)+P(X=1)+P(X=2)=e−4⋅400!+e−4⋅411!+e−4⋅422!≈0.018316+0.073263+0.146525 ...which simplifies to 0.238, rounded to the thousandths digit.

Karl and Fredo are basketball players who want to find out how they compare to their team in points per game. The mean amount of points per game and standard deviations for their team were calculated. Karl's z-score is 0.9. Fredo's z-score is −0.65. Which of the following statements are true about how Karl and Fredo compare to their team in points per game? Select all that apply.

Karl's average points per game is 0.9 standard deviations greater than his teammates' average points per game. Fredo's average points per game is closer to the team's mean than Karl's. The z-score is the number of standard deviations a data value is from the mean of the data set. Karl's average points per game is 0.9 standard deviations greater than his team mean. Fredo's average points per game is 0.65 standard deviations less than his team mean. But, since Fredo's z-score is −0.65, the distance from this to the mean is less than 0.9. since |−0.65|<0.9. So, Fredo's average points per game is closer to the team's mean than Karl's.

Suppose your favorite celebrity posts an average of 2.3 messages per day on a particular social media website. Find the probability that she posts exactly one message for any random day. Round your answer to three decimal places.

.231 Note that this is a Poisson probability. In this case, we want to find the probability of exactly 1 occurrence for a day. To determine the probability from a Poisson distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the POISSON.DIST function. 3. Next enter the values for the number of events and the mean or the expected number of occurrences per interval. In this case, enter 1 and 2.3, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.230595, which is 0.231 rounded to three decimal places.

Consider how the following scenario could be modeled with a binomial distribution, and answer the question that follows. 54.4% of tickets sold to a movie are sold with a popcorn coupon, and 45.6% are not. You want to calculate the probability of selling exactly 6 tickets with popcorn coupons out of 10 total tickets (or 6 successes in 10 trials). What value should you use for the parameter p?

0.544​ The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a movie ticket with a popcorn coupon, so p=0.544.

A database system assigns a 32-character ID to each record, where each character is either a number from 0 to 9 or a letter from A to F. Assume that each number or letter being selected is equally likely. Find the probability that at least 20 characters in the ID are numbers. Use Excel to find the probability. Round your answer to three decimal places.

0.578​ Note that this is a binomial probability. In this case, we want to find the probability of 20 to 32 successes, inclusive, where a success is a character being a number. This is the complement of the probability of 0 to 19 successes, inclusive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 19, 32, and 0.625, in that order. Enter 1 for Cumulative since this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.42192, which is 0.422 rounded to three decimal places. To find the probability of 20 to 32 successes, subtract this probability from 1. The probability is 1−0.422=0.578

Katie handles calls for a company's information technology help desk, either by resolving the issue herself or forwarding the call to the appropriate team. About 12.9% of the help desk calls need to be forwarded to another team. What is the probability that the first call Katie needs to forward to another team will be one of the first seven calls she receives that day? Use Excel to find the probability. Round your answer to three decimal places.

0.620 Use Excel to find the probability. Note that this is a cumulative probability. Let a success be a call that needs to be forwarded to another team. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the NEGBINOM.DIST function. 3. Next enter the values for the number of failures before the first success, the number of successes, and the probability of a success. In this case, since we want to determine the probability of at most seven trials, there are at most six failures before the first success. Thus, enter 6, 1, and 0.129, in that order. In the entry for a cumulative probability, enter 1 because this is a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.619699, which is 0.620 rounded to three decimal places

Random variable

a variable that is subject to change due to chance in events and describes the outcomes of a statistical experiment

Alex wants to test the reliability of "lie detector tests," or polygraph tests. He performs a polygraph test on a random sample of 12 individuals. If there is more than a 50% chance that the tests result in no false positives (that is, the test does not result in a true statement being recorded as a lie), Alex will conclude that the tests are reliable. If the probability of a lie detector test resulting in a false positive is 5.5%, what will Alex conclude? Use Excel to find the probability, rounding to three decimal places.

Alex will conclude that the test is reliable since the probability of no false positives is greater than 0.5. Note that this is a binomial probability. In this case, we want to find the probability of 0 successes, where a success is getting a false positive. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2. Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 0, 12, and 0.055, in that order. Enter 0 for Cumulative since this is not a cumulative probability. 4. Press OK. Excel should then display the probability. Here, the resulting probability is 0.507203, which is 0.507 rounded to three decimal places. This probability is greater than 0.5, so Alex will conclude that the test is reliable.

65% of the people in Missouri pass the driver's test on the first attempt. A group of 7 people took the test. Which of the following equations correctly calculate the probability that at least 3 in the group pass their driver's tests in their first atempt? Select all that apply. Remember: 65% = 0.65.

P(X≥3)=P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7) P(X≥3)=1−[P(X=2)+P(X=1)+P(X=0)] We need to find the probability of 3,4,5,6, and 7 friends passing their driver's tests in their first attempt. There are two different ways to find this probability: Sum the probabilities of each case. Find the probability of 0,1, and 2 friends passing their driver's tests in their first attempt. Add the probabilities and subtract the sum from 1 to use the cumulative rule.

A food processing plant fills snack-sized bags of crackers. The mean number of crackers in each bag is 22 and the standard deviation is 2. The factory supervisor selects one bag that contains 24 crackers. Which of the following statements is true?

The number of crackers in the supervisor's bag is 1 standard deviation to the right of the mean

Standard deviation

a number that measures how far data values or probability distributions are from their mean, denoted by a Greek letter (σ)