I have no idea where to even start.
Suppose heights of men are normally distributed with a mean 69.0 inches and standard deviation 2.8 inches. How many of a group of 1000 men would you expect to be between 70 and 72 inches tall? Round to the nearest whole number.
Suppose heights of men are normally distributed with a mean 69.0 inches and standard deviation 2.8 inches. For groups of 100 men, what would be the usual range of values for the mean height of these groups? Round the mean and standard deviation you use to the nearest tenth.
Suppose heights of men are normally distributed with a mean 69.0 inches and standard deviation 2.8 inches. A particular club has decided they will only allow the tallest 12% of men into their club. What is the shortest height that will be allowed in the club? Round to the nearest tenth of an inch.
To answer these questions step-by-step, let's break down each one:
1. To find the number of men between 70 and 72 inches tall in a group of 1000 men, we need to use the normal distribution and calculate the probability.
Step 1: Calculate the z-scores for the lower and upper limits.
z_lower = (70 - 69) / 2.8 = 0.36
z_upper = (72 - 69) / 2.8 = 1.07
Step 2: Look up the corresponding probabilities using a z-table (or use software like Excel or Python).
The probability for z_lower is 0.6449, and the probability for z_upper is 0.8577.
Step 3: Calculate the expected number of men:
Expected number = Probability of being in the range * Total number of men
Expected number = (0.8577 - 0.6449) * 1000 = 212.8
Rounded to the nearest whole number, we can expect approximately 213 men to be between 70 and 72 inches tall.
2. To determine the range of values for the mean height of groups of 100 men, we need to use the Central Limit Theorem.
Step 1: Calculate the standard deviation of the sample means, also known as the standard error.
Standard error = Standard deviation / √n
Standard error = 2.8 / √100 = 0.28
Step 2: Multiply the standard error by a z-value to find the range.
For a 95% confidence interval (which is commonly used), the z-value is approximately 1.96.
Range = Standard error * z-value
Range = 0.28 * 1.96 = 0.5496
Step 3: Calculate the usual range of values for the mean.
Usual range = Mean ± Range
Usual range = 69 ± 0.5496 = 68.5 to 69.5 (rounded to the nearest tenth)
So, the usual range of values for the mean height of groups of 100 men would be between 68.5 and 69.5 inches.
3. To determine the shortest height allowed in the club that only admits the tallest 12% of men, we need to calculate the corresponding z-value.
Step 1: Convert the percentile to a z-value using the standard normal distribution table (or use software).
The z-value corresponding to the 88th percentile (100% - 12%) is approximately 1.18.
Step 2: Calculate the raw score using the z-value formula.
Raw score = (z-value * standard deviation) + mean
Raw score = (1.18 * 2.8) + 69 = 72.104
Rounded to the nearest tenth of an inch, the shortest height allowed in the club is approximately 72.1 inches.
To answer these questions, we can use the properties of the normal distribution. Here's how you can calculate the answers step by step:
1. How many men would you expect to be between 70 and 72 inches tall in a group of 1000 men?
Step 1: Calculate the z-scores for the lower and upper limits.
z1 = (70 - 69) / 2.8
z2 = (72 - 69) / 2.8
Step 2: Find the probability of being below the upper limit using the z-score table or a calculator.
P(z < z2) = P(z < (72 - 69) / 2.8)
Step 3: Find the probability of being below the lower limit using the z-score table or a calculator.
P(z < z1) = P(z < (70 - 69) / 2.8)
Step 4: Subtract the two probabilities to find the probability between the two limits.
P(z1 < z < z2) = P(z < (72 - 69) / 2.8) - P(z < (70 - 69) / 2.8)
Step 5: Multiply the probability by the number of men (1000) to find the expected count.
Expected count = P(z1 < z < z2) * 1000
2. The usual range of values for the mean height of groups of 100 men.
Step 1: Calculate the standard error of the mean (SE).
SE = standard deviation / square root(group size)
SE = 2.8 / sqrt(100)
Step 2: Calculate the lower and upper limits of the usual range.
Lower limit = mean - 1.96 * SE
Upper limit = mean + 1.96 * SE
3. The shortest height that will be allowed in the club for the tallest 12% of men.
Step 1: Find the z-score that corresponds to the upper percentage.
z = inverse of cumulative distribution function (CDF) for the upper percentage.
z = inverse of CDF(0.88) (since 1 - 0.12 = 0.88)
Step 2: Calculate the height corresponding to the z-score.
Height = mean + (z * standard deviation)
Now, let's calculate the answers to these questions:
1. How many men would you expect to be between 70 and 72 inches tall in a group of 1000 men?
Calculate z1 and z2:
z1 = (70 - 69) / 2.8 = 0.357
z2 = (72 - 69) / 2.8 = 1.071
Use a z-score table or calculator to find the probabilities:
P(z < 0.357) = 0.6456
P(z < 1.071) = 0.8577
The probability between the two limits is:
P(z1 < z < z2) = P(z < 1.071) - P(z < 0.357) = 0.8577 - 0.6456 = 0.2121
Expected count = 0.2121 * 1000 = 212 (rounded to the nearest whole number)
Therefore, you would expect approximately 212 men to be between 70 and 72 inches tall in a group of 1000 men.
2. The usual range of values for the mean height of groups of 100 men.
SE = 2.8 / sqrt(100) = 0.28
Lower limit = 69 - 1.96 * 0.28 = 68.55 (rounded to the nearest tenth)
Upper limit = 69 + 1.96 * 0.28 = 69.45 (rounded to the nearest tenth)
The usual range for the mean height of groups of 100 men is approximately 68.6 to 69.4 inches.
3. The shortest height that will be allowed in the club for the tallest 12% of men.
Find the z-score for the upper percentage:
z = inverse of CDF(0.88)
z = inverse of CDF(0.88) ≈ 1.18
Height = 69 + (1.18 * 2.8) ≈ 72.3 (rounded to the nearest tenth)
Therefore, the shortest height that will be allowed in the club is approximately 72.3 inches.
Z = (score-mean)SD
Look up Z scores to get probability. Multiply by 1000.
DK second.
Third problem use table for .12 to find Z score. Insert Z value in equation above.