a. To determine if the sampling distribution of the sample mean is normally distributed, we need to check if the sample size is large enough for the Central Limit Theorem to apply. The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean becomes more and more normally distributed, regardless of the shape of the population distribution.
In this case, n = 16 and n = 41. According to the Central Limit Theorem, a sample size of 30 or larger is typically considered large enough for the sampling distribution of the sample mean to be considered approximately normal.
Therefore, with n = 16 and n = 41, we can conclude that the sampling distribution of the sample mean is normally distributed for both sample sizes.
b. The answer is: Yes, both the sample means will have a normal distribution.
c. To calculate the probability that the sample mean falls between 68 and 71 with a normally distributed sampling distribution, we need to first calculate the z-scores for each value using the formula:
z = (x - μ) / (σ / sqrt(n))
For x = 68, μ = 68, σ = 5.7, and n = 16:
z1 = (68 - 68) / (5.7 / sqrt(16)) = 0
For x = 71:
z2 = (71 - 68) / (5.7 / sqrt(16)) = 1.403
Next, we need to find the corresponding probabilities from the z-table. The probability that the sample mean falls between 68 and 71 is given by the difference in these probabilities:
P(68 < x < 71) = P(0 < z < 1.403)
Using the z-table, we find that P(0 < z < 1.403) is approximately 0.9192.
Therefore, the probability that the sample mean falls between 68 and 71 is approximately 0.9192.
d. Following similar steps as in part c, we calculate the z-scores for x = 68 and x = 71 with n = 41:
For x = 68:
z1 = (68 - 68) / (5.7 / sqrt(41)) = 0
For x = 71:
z2 = (71 - 68) / (5.7 / sqrt(41)) = 1.929
Using the z-table, we find that P(0 < z < 1.929) is approximately 0.9738.
Therefore, the probability that the sample mean falls between 68 and 71 is approximately 0.9738.
A random sample is drawn from a population with mean μ = 68 and standard deviation σ = 5.7. [You may find it useful to reference the z table.]
a. Is the sampling distribution of the sample mean with n = 16 and n = 41 normally distributed? (Round the standard error to 3 decimal places.)
b. Can you conclude that the sampling distribution of the sample mean is normally distributed for both sample sizes?
multiple choice 1
Yes, both the sample means will have a normal distribution.
No, both the sample means will not have a normal distribution.
No, only the sample mean with n = 16 will have a normal distribution.
No, only the sample mean with n = 41 will have a normal distribution.
c. If the sampling distribution of the sample mean is normally distributed with n = 16, then calculate the probability that the sample mean falls between 68 and 71. (If appropriate, round final answer to 4 decimal places.)
multiple choice 2
We cannot assume that the sampling distribution of the sample mean is normally distributed.
We can assume that the sampling distribution of the sample mean is normally distributed and the probability that the sample mean falls between 68 and 71 is
d. If the sampling distribution of the sample mean is normally distributed with n = 41, then calculate the probability that the sample mean falls between 68 and 71. (If appropriate, round final answer to 4 decimal places.)
multiple choice 3
We cannot assume that the sampling distribution of the sample mean is normally distributed.
We can assume that the sampling distribution of the sample mean is normally distributed and the probability that the sample mean falls between 68 and 71 is
1 answer