An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 140 lb and 181 lb. The new population of pilots has normally distributed weights with a mean of 146 lb and a standard deviation of 28.9 lb.
A) If a pilot is randomly selected, find the probability that his weight is between 140 and 181 lb.
B) If 32 different pilots are randomly selected, find the probability that their mean weight is between 140 and 181 lb
A) Z = (score-mean)/SD
B) Z = (score-mean)/SEm
SEm = SD/√n
For both, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
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A) To find the probability that a randomly selected pilot's weight is between 140 and 181 pounds, we need to calculate the Z-scores for both weight values and then find the area under the normal distribution curve between these Z-scores.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
X = pilot's weight
μ = mean weight
σ = standard deviation
For the lower weight limit:
Z1 = (140 - 146) / 28.9
For the upper weight limit:
Z2 = (181 - 146) / 28.9
Using a Z-table or a statistical software, we can find the area under the curve for each Z-score.
Let's calculate the Z-scores first:
Z1 = (140 - 146) / 28.9
Z1 = -0.207
Z2 = (181 - 146) / 28.9
Z2 = 1.210
Now, we need to find the area between these two Z-scores. We can use a Z-table or a statistical software to find these probabilities.
The probability of a pilot's weight being between 140 and 181 pounds is the difference between the cumulative probabilities associated with Z1 and Z2:
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
You can use a Z-table or a statistical software to find these probabilities by looking up the Z-scores.
B) To find the probability that the mean weight of 32 randomly selected pilots falls between 140 and 181 pounds, we need to use the Central Limit Theorem. According to the Central Limit Theorem, if a random sample is drawn from a population with any distribution, the sample mean will be approximately normally distributed if the sample size is large enough.
The mean of the sample mean will be equal to the population mean, and the standard deviation of the sample mean, also known as the standard error of the mean (SE), will be equal to the population standard deviation divided by the square root of the sample size.
The formula to calculate the standard error of the mean is given by:
SE = σ / √n
Where:
SE = standard error of the mean
σ = population standard deviation
n = sample size
In this case, the population standard deviation (σ) is 28.9 pounds, and the sample size (n) is 32 pilots.
SE = 28.9 / √32
Once we have the standard error, we can use Z-scores to find the probability that the sample mean falls between 140 and 181 pounds.
To do this, we can calculate the Z-score for each weight value and use the Z-table or a statistical software to find the area under the curve between these Z-scores.
Z1 = (140 - 146) / (28.9 / √32)
Z2 = (181 - 146) / (28.9 / √32)
Using the Z-scores, we can find the probability of the sample mean falling between these values. Again, we can use a Z-table or a statistical software to find these probabilities by looking up the Z-scores.
To solve these probability questions, we will use the concept of the standard normal distribution. To work with the standard normal distribution, we need to standardize the given values using z-scores.
The formula for calculating the z-score of a given value is:
z = (x - μ) / σ
Where:
- z is the standard score (z-score) of a value
- x is the given value
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
A) To find the probability that a pilot's weight is between 140 and 181 lb, we will calculate the z-scores for the lower and upper limits and then use a standard normal distribution table (or a calculator) to find the corresponding probabilities.
For the lower limit (140 lb):
z = (140 - 146) / 28.9 = -0.207
For the upper limit (181 lb):
z = (181 - 146) / 28.9 = 1.208
Now, we can use the standard normal distribution table to find the probabilities associated with these z-scores. Subtracting the cumulative probability for the lower limit from the cumulative probability for the upper limit will give us the desired probability:
P(140 ≤ x ≤ 181) = P(z ≤ 1.208) - P(z ≤ -0.207)
B) To find the probability that the mean weight of 32 randomly selected pilots is between 140 and 181 lb, we will use the Central Limit Theorem. According to this theorem, the distribution of the sample means will approach a normal distribution, regardless of the shape of the original population distribution.
The mean of the sample means will still be the same as the population mean (146 lb), but the standard deviation of the sample means, also known as the standard error of the mean, can be calculated as σ / sqrt(n), where n is the sample size.
In this case, n = 32, so the standard error of the mean is:
standard error = 28.9 / sqrt(32) ≈ 5.10
Now, we will standardize the lower and upper limits using the formula mentioned earlier:
For 140 lb:
z = (140 - 146) / 5.10 ≈ -1.18
For 181 lb:
z = (181 - 146) / 5.10 ≈ 6.86
Next, we will use the standard normal distribution table to find the probabilities associated with these z-scores, just like in part A:
P(mean weight between 140 and 181 lb) = P(z ≤ 6.86) - P(z ≤ -1.18)
Remember to consult a standard normal distribution table, a statistical software, or an online calculator to find the precise probabilities corresponding to the calculated z-scores.