7. The random variable X is distributed normally with a mean of 12.46 and variance of 13.11. You collect a random sample of size 37.
a. What is the probability that your sample mean is between 12 and 13?
b. What is the probability that a single observation is between 12 and 13?
c. What is the probability that all of your observations are between 12 and 13?
a. Z = (score-mean)/SEm
SEm = SD/√n
SD = √variance
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
b. Z = (score-mean)/SD
Use same table.
c. I'm not sure about this one.
To find the probabilities, we will make use of the properties of the normal distribution. The mean and variance provided allow us to assume that the sample mean (a) and a single observation (b) follow a normal distribution.
a. Probability that the sample mean is between 12 and 13:
The sample mean follows a normal distribution with the same mean as the population mean (12.46) and a variance equal to the population variance divided by the sample size (13.11 / 37).
First, we need to calculate the standard deviation of the sample mean, which is the square root of the variance (13.11 / 37).
Standard deviation = √(13.11 / 37) ≈ 0.3309
Next, we can calculate the z-scores for the lower and upper bounds of the interval:
Lower bound z-score = (12 - 12.46) / 0.3309 ≈ -1.39
Upper bound z-score = (13 - 12.46) / 0.3309 ≈ 1.64
Using a z-table or a statistical software, we can find the probabilities associated with these z-scores:
Probability (sample mean between 12 and 13) = Probability (-1.39 < z < 1.64)
By looking up these z-scores in the z-table, we find the probabilities associated with them. Subtracting the lower probability from the upper probability gives us the answer.
b. Probability that a single observation is between 12 and 13:
Since a single observation is assumed to follow a normal distribution with the given mean and variance, we can use the z-score formula to calculate the z-score associated with each boundary.
Lower bound z-score = (12 - 12.46) / √13.11 ≈ -0.31
Upper bound z-score = (13 - 12.46) / √13.11 ≈ 0.27
Again, using a z-table, we can find the probabilities associated with these z-scores:
Probability (single observation between 12 and 13) = Probability (-0.31 < z < 0.27)
c. Probability that all of the observations are between 12 and 13:
Since each observation is assumed to be independent and normally distributed, the probability that each individual observation falls within the given range can be calculated using the z-score formula, as in part b.
To find the probability that all of the observations fall within the range, we multiply the individual probabilities together:
Probability (all observations between 12 and 13) = Probability (observation 1 between 12 and 13) * Probability (observation 2 between 12 and 13) * ... * Probability (observation 37 between 12 and 13)