Question

Q1. The length of human pregnancies are approximately normally distributed with a mean of μ=266 days and standard deviation σ=16 days. What percent of pregnancies last between 240 and 280​days? Give your answer to the nearest​ 1%. ____%

Q2. According to data from the U.S. Geological​ Survey, the magnitude of earthquakes in California since 1900 that measure 0.1 or higher on the Richter scale is approximately normally distributed with a mean of μ=6.2 and standard deviation σ=0.5. Determine the 15th percentile of the magnitude of earthquakes in California. Give 1 decimal place in your answer. ______

Q3. What is the probability in a family of four children that there are two boys and two​ girls? ____

What is the probability in a family of eight children that there are four girls and four​ boys? ____

Q4. What is the probability of rolling exactly one 6 with four​ die? _____

What is the probability of rolling at least one 6 with four​ die?   ______

Answer 1

Answer 2

Q1. To find the percentage of pregnancies that last between 240 and 280 days, we need to find the area under the normal distribution curve between these two values. We can do this by standardizing the values using the formula z = (x - μ) / σ, where x is the value we're interested in, μ is the mean, and σ is the standard deviation.

For x = 240 days, z = (240 - 266) / 16 = -1.625. For x = 280 days, z = (280 - 266) / 16 = 0.875. Using a standard normal distribution table or calculator, we can find that the area under the curve between z = -1.625 and z = 0.875 is approximately 81%. Therefore, about 81% of pregnancies last between 240 and 280 days.

Q2. To find the 15th percentile of earthquake magnitudes in California, we need to find the value of x such that P(X ≤ x) = 0.15, where X is the magnitude of earthquakes. We can use the same formula as in Q1 to standardize the value of x: z = (x - μ) / σ.

Substituting the given values, we have z = (x - 6.2) / 0.5. We want to find the value of x that corresponds to the 15th percentile, which means that 15% of earthquakes have a magnitude of x or lower. Using a standard normal distribution table or calculator, we can find that the z-score that corresponds to the 15th percentile is approximately -1.04.

Substituting this value into the formula for z, we get -1.04 = (x - 6.2) / 0.5. Solving for x, we get x = 5.18 (rounded to one decimal place). Therefore, the 15th percentile of earthquake magnitudes in California is 5.2.

Q3. To find the probability of having two boys and two girls in a family of four children, we can use the binomial distribution formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where X is the random variable representing the number of boys, n is the sample size (number of children), k is the number of successes (in this case, two boys), and p is the probability of a success (in this case, the probability of having a boy, which is 0.5 assuming an equal chance of having a boy or girl).

Substituting the values, we have P(X = 2) = (4 choose 2) * 0.5^2 * 0.5^2 = 0.375. Therefore, the probability of having two boys and two girls in a family of four children is 0.375.

To find the probability of having four boys and four girls in a family of eight children, we can use the same formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k).

Substituting the values, we have P(X = 4) = (8 choose 4) * 0.5^4 * 0.5^4 = 0.1367 (rounded to four decimal places). Therefore, the probability of having four boys and four girls in a family of eight children is 0.1367.

Q4. To find the probability of rolling exactly one 6 with four dice, we can use the binomial distribution formula: P(X = k) = (n choose