Question 1

The time that it takes an Olympic runner to finish the 100-meter dash is approximately normal with a mean of 9.83 seconds with a standard deviation of 0.06 seconds. What is the probability it would take The runner less than 10.01 seconds to finish the race? Give your answer as a percentage.

Accepted Answer

To find the probability, we first calculate the Z-score using the formula Z = (X - μ) / σ, where X is the value of interest (10.01 seconds), μ is the mean (9.83 seconds), and σ is the standard deviation (0.06 seconds). Plugging in the values, we get Z = (10.01 - 9.83) / 0.06 = 3. This Z-score corresponds to a probability of 99.85% when looking up in a standard normal distribution table or using a calculator, indicating the probability of a runner finishing in less than 10.01 seconds.

Question 2

The graph below is of a normal curve. The shaded portion of the graph represents 68 % of the total area under the curve. According to the graph, what is the value of the 99.85^th percentile P_99.85 ?

Accepted Answer

The answer of The graph below is of a normal...

Question 3

A local church lottery states that the probability that any ticket wins a prize is 36%. Suppose that you are generous and purchase 1225 tickets, what is the probability that more than 441 of the tickets result In a prize? Give your answer as a percentage.

Accepted Answer

The answer of A local church lottery states that the...

Question 4

The height of an 18-year old student at your university has an approximate normal distribution with a mean of 68 inches and a standard deviation of 2.5 inches. What is the first quartile of height in this Distribution? Round your answer to two decimal places.

Accepted Answer

The first quartile (Q1) in a normal distribution can be found using the Z-score table. The first quartile corresponds to the 25th percentile, which has a Z-score of approximately -0.675. Using the formula: X = μ + Zσ, where μ is the mean, σ is the standard deviation, and Z is the Z-score, we calculate the first quartile as follows: X = 68 + (-0.675)(2.5) = 66.31 inches.

Question 5

A normal distribution has a mean of and a standard deviation of . Suppose that you know that 2.5^th percentile P₂₅=8 and the 84^th percentile P₈₄=29 . Find the mean of this normal distribution.

Accepted Answer

The answer of A normal distribution has a mean of...

Question 6

An honest six-sided die is tossed a total of n = 324 times and the number of times that the die lands on an even number (2, 4, or 6) is recorded. What is the probability that between 144 and 171 of the tosses Result in an even number? Give your answer as a percentage.

Accepted Answer

The answer of An honest six-sided die is tossed a...

Question 7

The time that it takes an Olympic runner to finish the 100-meter dash is approximately normal with a mean of 9.83 seconds with a standard deviation of 0.06 seconds. What is the probability it would take The runner more than 9.89 seconds to finish the race? Give your answer as a percentage.

Accepted Answer

The answer of The time that it takes an Olympic...

Question 8

The graph below is of a normal curve. The shaded portion of the graph represents 68% of the total area under the curve. According to the graph, what is the value of the 97.5th percentile P97.5 ?

Accepted Answer

The answer of The graph below is of a normal...

Question 9

The actual number of hours that a candle will burn has an approximate normal distribution with a mean of 18 hours and a standard deviation of 1.2 hours. What is the probability that the candle will burn for Less than 15.6 hours? Give your answer as a percentage.

Accepted Answer

To find the probability, we calculate the z-score: z = (15.6 - 18) / 1.2 = -2. The probability of a z-score less than -2 is approximately 2.5% in a standard normal distribution.

Question 10

The time that it takes a student to walk from his dorm room to his math classroom has an approximate normal distribution with a mean of 8 minutes and a standard deviation of 0.4 minutes. What is the Probability that the student will take between 8.4 and 9.2 minutes to walk to class? Give your answer As a percentage.

Accepted Answer

The Z-scores for 8.4 and 9.2 minutes are 1 and 3, respectively. The probability of a Z-score between 1 and 3 is approximately 15.85%.

Question 11

A local church lottery states that the probability that any ticket wins a prize is 36%. Suppose that you are generous and purchase 1225 tickets, what is the probability that less than 441 of the tickets result in A prize? Give your answer as a percentage.

Accepted Answer

Given the large number of tickets (1225) and a specific probability (36%), the scenario can be approximated using the normal distribution due to the Central Limit Theorem. The mean (μ) of the distribution of winning tickets is given by n*p, where n is the number of trials (1225 tickets) and p is the probability of success (0.36). Thus, μ = 1225 * 0.36 = 441. The standard deviation (σ) is given by sqrt(n*p*(1-p)), which calculates to sqrt(1225*0.36*0.64) ≈ 18.3.To find the probability of winning less than 441 tickets, we look for the probability of getting results below the mean in a normal distribution, which is essentially asking for the probability of a result less than or equal to the mean. In a perfectly symmetrical distribution, this would be 50%. However, the question asks for the probability of less than 441, not less than or equal to. In practical terms, for a large sample like this, the difference between less than 441 and less than or equal to 441 is negligible. However, the options provided do not account for this specific scenario accurately, as they seem to suggest a specific calculation or approximation that doesn't directly apply to the information given. The correct approach using normal approximation would involve calculating a z-score and then looking up that value in a standard normal distribution table or using a calculator. Since the exact calculation or approximation isn't provided in the options, and given the nature of the question, the most accurate answer with the information provided is "None of the above."

Question 12

Which of the following scenarios would it not be appropriate to model with a normal distribution?

Accepted Answer

The number of fingers on the left hand of an 18-year female student at your university is a discrete variable and typically does not vary in a population (most people have 5 fingers), making it inappropriate for modeling with a normal distribution. The other options involve continuous variables that can be reasonably assumed to follow a normal distribution due to natural variation in the population.

Question 13

The actual number of hours that a candle will burn has an approximate normal distribution with a mean of 18 hours and a standard deviation of 1.2 hours. What is the probability that the candle will burn for Less than 14.4 hours? Give your answer as a percentage.

Accepted Answer

The answer of The actual number of hours that a...

Question 14

A normal distribution has a mean of and a standard deviation of . Suppose that you know that 16^th percentile P₁₆=23 and the 97.5^th percentile P_97.5=62 . Find the mean of this normal distribution.

Accepted Answer

The answer of A normal distribution has a mean of...

Question 15

The time that it takes a student to walk from his dorm room to his math classroom has an approximate normal distribution with a mean of 8 minutes and a standard deviation of 0.4 minutes. What is the Probability that the student will take between 8.0 and 9.2 minutes to walk to class? Give your answer As a percentage.

Accepted Answer

The probability that the student will take between 8.0 and 9.2 minutes to walk to class can be calculated using the normal distribution formula:P(8.0 < X < 9.2) = P(Z < (9.2 - 8) / 0.4) - P(Z < (8.0 - 8) / 0.4)= P(Z < 3) - P(Z < 0)= 0.9987 - 0.5= 0.4987Therefore, the probability that the student will take between 8.0 and 9.2 minutes to walk to class is 0.4987, or 49.85%.

Question 16

Which of the following scenarios would it be appropriate to model with a normal distribution?

Accepted Answer

The height of an 18-year-old male student at a university is a continuous variable that can take on a range of values and is likely to follow a normal distribution pattern, where most individuals' heights cluster around the mean, with fewer individuals being extremely tall or extremely short.

Question 17

In a normal distribution with a mean of 30 , the number 45 has a standardized value of 1.2 . What is the 16^th percentile P₁₆ of this normal distribution?

Accepted Answer

The answer of In a normal distribution with a mean...

Question 18

In a normal distribution with a standard deviation of 8 , the number 25 has a standardized value of -2.5 . What is the 84^th percentile P₈₄ of this normal distribution?

Accepted Answer

The answer of In a normal distribution with a standard...

Question 19

The height of an 18-year old student at your university has an approximate normal distribution with a mean of 68 inches and a standard deviation of 2.5 inches. What is the third quartile of height in this Distribution? Round your answer to two decimal places.

Accepted Answer

The third quartile (Q3) in a normal distribution can be found using the Z-score table. The third quartile corresponds to the 75th percentile, which has a Z-score of approximately 0.674. Using the formula for converting Z-scores to raw scores, $X = \mu + Z\sigma$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $Z$ is the Z-score, we get $X = 68 + 0.674(2.5) = 68 + 1.685 = 69.69$ inches.

Question 20

An honest six-sided die is tossed a total of n = 324 times and the number of times that the die lands on an even number (2, 4, or 6) is recorded. What is the probability that between 162 and 171 of the tosses Result in an even number? Give your answer as a percentage.

Accepted Answer