Question 1

A certain rare form of cancer occurs in 37 children in a million, so its probability is 0.000037. In the city of Normalville there are 74,090 children. A Poisson distribution will be used to approximate the probability that
The number of cases of the disease in Normalville children is more than 2. Find the mean of the appropriate
Poisson distribution (the mean number of cases in groups of 74,090 children).

Accepted Answer

The mean of the Poisson distribution is calculated by multiplying the probability of the event (0.000037) by the number of trials (74,090 children), which gives 2.74, approximately 2.7.

Question 2

The following table describes the results of roadworthiness tests of Ford Focus cars that are three years old (based on data from the Department of Transportation). The random variable x represents the number of cars
That failed among six that were tested for roadworthiness: $$\begin{array} { | c | c | } 
\hline x & P ( x ) \
\hline 0 & 0.377 \
\hline 1 & 0.399 \
\hline 2 & 0.176 \
\hline 3 & 0.041 \
\hline 4 & 0.005 \
\hline 5 & 0 + \
\hline 6 & 0 + \
\hline \hline
\end{array}$$ Find the probability of getting three or more cars that fail among six cars tested.

Accepted Answer

The answer of The following table describes the results of...

Question 3

A tennis player makes a successful first serve 51% of the time. If she serves 9 times, what is the probability that she gets exactly 3 successful first serves in? Assume that each serve is independent of the others.

Accepted Answer

The probability of exactly 3 successful serves out of 9 can be calculated using the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n=9, k=3, and p=0.51. This results in approximately 0.154.

Question 4

Determine whether the following is a probability distribution. If not, identify the requirement that is not
satisfied. $$\begin{array} { | c | c | } 
\hline x & P ( x ) \
\hline 0 & 0.1296 \
\hline 1 & 0.3456 \
\hline 2 & 0.3456 \
\hline 3 & 0.1536 \
\hline 4 & 0.0256 \
\hline
\end{array}$$

Accepted Answer

The answer of Determine whether the following is a probability...

Question 5

Identify the given random variable as being discrete or continuous. The cost of a randomly selected orange.

Accepted Answer

The cost of an orange can take on any value within a certain range, so it is a continuous random variable.

Question 6

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied. 
$\begin{array} { | c | c | } 
\hline x & P ( x ) \
\hline 0 & 0.109 \
\hline 1 & 0.208 \
\hline 2 & 0.246 \
\hline 3 & 0.159 \
\hline 4 & 0.096 \
\hline 5 & 0.228 \
\hline
\end{array}$

Accepted Answer

The answer of Determine whether the following is a probability...

Question 7

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 53 times, keeping track of the "fives" rolled.

Accepted Answer

Rolling a die 53 times and tracking "fives" is a binomial distribution because each roll is independent, there are two outcomes (rolling a five or not), a fixed number of trials, and a constant probability of rolling a five.

Question 8

The following table describes the results of roadworthiness tests of Ford Focus cars that are three years old (based on data from the Department of Transportation). The random variable x represents the number of cars
That failed among six that were tested for roadworthiness: $$\begin{array} { | c | c | } 
\hline x & P ( x ) \
\hline 0 & 0.377 \
\hline 1 & 0.399 \
\hline 2 & 0.176 \
\hline 3 & 0.041 \
\hline 4 & 0.005 \
\hline 5 & 0 + \
\hline 6 & 0 + \
\hline
\end{array}$$ Is the probability of getting three or more cars that fail among six cars tested significant, determined by a cutoff value of
0)05?

Accepted Answer

To determine if the probability of getting three or more cars that fail among six cars tested is significant, we need to calculate the probability of getting three or more failures and compare it to the cutoff value of 0.05.

P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)
P(X ≥ 3) = 0.041 + 0.005 + 0 + 0
P(X ≥ 3) = 0.046

Since 0.046 is less than 0.05, we can conclude that the probability of getting three or more cars that fail among six cars tested is not significant at the 0.05 cutoff value. Therefore, the answer is A) Yes.

Question 9

A survey for brand recognition is done and it is determined that 68% of consumers have heard of Dull Computer Company. A survey of 800 randomly selected consumers is to be conducted. For such groups of 800,
Would it be significant to get 634 consumers who recognize the Dull Computer Company name? Consider as
Significant any result that differs from the mean by more than 2 standard deviations. That is, significant values
Are either less than $\mu - 2 \sigma$ or greater than $\mu + 2 \sigma$

Accepted Answer

To determine whether it would be significant to get 634 consumers who recognize the Dull Computer Company name, we need to calculate the mean and standard deviation of the sample proportion.

The mean of the sample proportion is the same as the population proportion, which is 0.68.

The standard deviation of the sample proportion can be calculated using the formula:

$\sigma = \sqrt{\frac{p(1-p)}{n}}$

where p is the population proportion (0.68), and n is the sample size (800).

Plugging in the values, we get:

$\sigma = \sqrt{\frac{0.68(1-0.68)}{800}} = 0.022$

To find the value that differs from the mean by 2 standard deviations, we need to multiply the standard deviation by 2 and add/subtract the result to/from the mean:

$\mu + 2\sigma = 0.68 + 2(0.022) = 0.724$

$\mu - 2\sigma = 0.68 - 2(0.022) = 0.636$

So any result that is greater than 0.724 or less than 0.636 would be considered significant.

To find out if 634 is significant, we need to calculate the sample proportion:

$\hat{p} = \frac{x}{n} = \frac{634}{800} = 0.7925$

This result is greater than the mean (0.68), but is it greater than 2 standard deviations away from the mean?

To find out, we can calculate the z-score:

$z = \frac{\hat{p}-p}{\sigma} = \frac{0.7925-0.68}{0.022} = 5.114$

A z-score of 5.114 is definitely greater than 2, which means that a result of 634 would be considered significant. Therefore, the answer is A) Yes.

Question 10

Based on a Comcast survey, there is a 0.8 probability that a randomly selected adult will watch prime-time TV live, instead of online, on DVR, etc. Assume that seven adults are randomly selected. Find the probability that
Fewer than three of the selected adults watch prime-time live.

Accepted Answer

The answer of Based on a Comcast survey, there is...

Question 11

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Spinning a roulette wheel 9 times, keeping track of the occurrences of a winning number of "16."

Accepted Answer

The procedure results in a binomial distribution because there are only two possible outcomes (win or lose) for each trial, the trials are independent, and there are a fixed number of trials (9).

Question 12

Find the standard deviation, σ, for the binomial distribution with n = 38 and p = 0.4.

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula $\sigma = \sqrt{n \cdot p \cdot (1-p)}$. Substituting the given values, $\sigma = \sqrt{38 \cdot 0.4 \cdot 0.6} = \sqrt{9.12} \approx 3.02$.

Question 13

A company manufactures batteries in batches of 6 and there is a 3% rate of defects. Find the mean number of defects per batch.

Accepted Answer

The mean number of defects per batch is calculated by multiplying the batch size (6) by the defect rate (3%), which equals 0.18 defects per batch, rounded to 0.2 defects.

Question 14

According to a college survey, 22% of all students work full time. Find the mean for the number of students who work full time in samples of size 16.

Accepted Answer

The answer of According to a college survey, 22% of...

Question 15

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test? Round to three
Decimal places.

Accepted Answer

The probability of getting any one question correct by guessing is 1/2. Since there are 10 questions, the number of ways to get exactly 6 questions correct is given by the binomial distribution with n = 10 and p = 1/2:
P(X = 6) = (10 choose 6) * (1/2)^6 * (1/2)^4 = 0.205
Similarly, we can find the probabilities of getting exactly 7, 8, 9, or 10 questions correct and add them together, since passing the test requires getting at least 6 questions correct:
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
P(X ≥ 6) = 0.205 + 0.246 + 0.205 + 0.102 + 0.028 = 0.786
Rounding to three decimal places, we get 0.786, which matches choice B.

Question 16

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 26 times, keeping track of the numbers that are rolled.

Accepted Answer

A binomial distribution only has two possible outcomes for each trial (such as success or failure) and this procedure has six possible outcomes (the numbers on the die). Therefore, it cannot result in a binomial distribution.

Question 17

In a certain college, 33% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that no more than 6 belong to an ethnic minority? Round to
Three decimal places.

Accepted Answer

This is a binomial probability problem where n = 10, p = 0.33, and we want P(X ≤ 6). Using the binomial probability formula or a calculator, the probability is approximately 0.982.

Question 18

The Acme Candy Company claims that $60 \%$ of the jawbreakers it produces weigh more than $0.4$ ounces. Suppose that 800 jawbreakers are selected at random from the production lines. Would it be significant for this sample of 800 to contain 494 jawbreakers that weigh more than $0.4$ ounces? Consider as significant any result that differs from the mean by more than 2 standard deviations. That is, significant values are either less than $\mu$ $2 \sigma$ or greater than $\mu + 2 \sigma$.

Accepted Answer

The expected number of jawbreakers weighing more than 0.4 ounces is $0.6 \times 800 = 480$. The standard deviation is $\sqrt{800 \times 0.6 \times 0.4} = \sqrt{192} \approx 13.86$. The range for non-significant results is $480 \pm 2 \times 13.86$, which is approximately $452.28$ to $507.72$. Since 494 is within this range, it is not significant.

Question 19

In one town, the number of burglaries in a week has a Poisson distribution with a mean of 1.9 burglaries. Find the probability that in a randomly selected week the number of burglaries is at least three. Use the Poisson
Distribution to find the indicated probability.

Accepted Answer

We can use the Poisson distribution formula to solve this problem:
P(X ≥ 3) = 1 - P(X < 3)
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
P(X = k) = (e^(-λ) * λ^k) / k!
where λ = 1.9 (mean)
P(X = 0) = (e^-1.9) * 1.9^0 / 0! = 0.1499
P(X = 1) = (e^-1.9) * 1.9^1 / 1! = 0.2849
P(X = 2) = (e^-1.9) * 1.9^2 / 2! = 0.2709
P(X < 3) = 0.1499 + 0.2849 + 0.2709 = 0.7057
P(X ≥ 3) = 1 - 0.7057 = 0.2943 ≈ 0.2963 (rounded to four decimal places)
Therefore, the answer is D.

Question 20

Use the given values of n = 1205 and p = 0.98 to find the minimum value that is not significantly low, $\mu - 2 \sigma ,$ and the maximum value that is not significantly high, $\mu + 2 \sigma$ Round your answer to the nearest hundredth unless
Otherwise noted.

Accepted Answer

The answer of Use the given values of n =...

Quiz 5: Discrete Probability Distributions

A tennis player makes a successful first serve 51% of the time. If she serves 9 times, what is the probability that she gets exactly 3 successful first serves in? Assume that each serve is independent of the others.

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied. $\begin{array} { | c | c | } \hline x & P ( x ) \\\hline 0 & 0.1296 \\\hline 1 & 0.3456 \\\hline 2 & 0.3456 \\\hline 3 & 0.1536 \\\hline 4 & 0.0256 \\\hline\end{array}$

Identify the given random variable as being discrete or continuous. The cost of a randomly selected orange.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 53 times, keeping track of the "fives" rolled.

Based on a Comcast survey, there is a 0.8 probability that a randomly selected adult will watch prime-time TV live, instead of online, on DVR, etc. Assume that seven adults are randomly selected. Find the probability that Fewer than three of the selected adults watch prime-time live.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Spinning a roulette wheel 9 times, keeping track of the occurrences of a winning number of "16."

Find the standard deviation, σ, for the binomial distribution with n = 38 and p = 0.4.

A company manufactures batteries in batches of 6 and there is a 3% rate of defects. Find the mean number of defects per batch.

According to a college survey, 22% of all students work full time. Find the mean for the number of students who work full time in samples of size 16.

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test? Round to three Decimal places.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 26 times, keeping track of the numbers that are rolled.

In a certain college, 33% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that no more than 6 belong to an ethnic minority? Round to Three decimal places.

In one town, the number of burglaries in a week has a Poisson distribution with a mean of 1.9 burglaries. Find the probability that in a randomly selected week the number of burglaries is at least three. Use the Poisson Distribution to find the indicated probability.

Use the given values of n = 1205 and p = 0.98 to find the minimum value that is not significantly low, $\mu - 2 \sigma ,$ and the maximum value that is not significantly high, $\mu + 2 \sigma$ Round your answer to the nearest hundredth unless Otherwise noted.

Quiz 5: Discrete Probability Distributions

A tennis player makes a successful first serve 51% of the time. If she serves 9 times, what is the probability that she gets exactly 3 successful first serves in? Assume that each serve is independent of the others.

Identify the given random variable as being discrete or continuous. The cost of a randomly selected orange.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 53 times, keeping track of the "fives" rolled.

Based on a Comcast survey, there is a 0.8 probability that a randomly selected adult will watch prime-time TV live, instead of online, on DVR, etc. Assume that seven adults are randomly selected. Find the probability that Fewer than three of the selected adults watch prime-time live.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Spinning a roulette wheel 9 times, keeping track of the occurrences of a winning number of "16."

Find the standard deviation, σ, for the binomial distribution with n = 38 and p = 0.4.

A company manufactures batteries in batches of 6 and there is a 3% rate of defects. Find the mean number of defects per batch.

According to a college survey, 22% of all students work full time. Find the mean for the number of students who work full time in samples of size 16.

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test? Round to three Decimal places.

Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 26 times, keeping track of the numbers that are rolled.

In a certain college, 33% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that no more than 6 belong to an ethnic minority? Round to Three decimal places.

In one town, the number of burglaries in a week has a Poisson distribution with a mean of 1.9 burglaries. Find the probability that in a randomly selected week the number of burglaries is at least three. Use the Poisson Distribution to find the indicated probability.

Use the given values of n = 1205 and p = 0.98 to find the minimum value that is not significantly low, μ−2σ,\mu - 2 \sigma ,μ−2σ, and the maximum value that is not significantly high, μ+2σ\mu + 2 \sigmaμ+2σ Round your answer to the nearest hundredth unless Otherwise noted.

Use the given values of n = 1205 and p = 0.98 to find the minimum value that is not significantly low, $\mu - 2 \sigma ,$ and the maximum value that is not significantly high, $\mu + 2 \sigma$ Round your answer to the nearest hundredth unless Otherwise noted.