a.
For n = 17:
Expected Value = p = 0.27
Standard Error = sqrt(p(1-p)/n) = sqrt(0.27(1-0.27)/17) ≈ 0.0624
For n = 65:
Expected Value = p = 0.27
Standard Error = sqrt(p(1-p)/n) = sqrt(0.27(1-0.27)/65) ≈ 0.0480
b. No, only the sample proportion with n = 65 will have a normal distribution.
c. To calculate the probability that the sample proportion is between 0.25 and 0.27 with n = 17, we can calculate the z-scores for the lower and upper bounds and use the z-table.
Lower Bound z-score = (0.25 - 0.27) / 0.0624 ≈ -0.0321
Upper Bound z-score = (0.27 - 0.27) / 0.0624 = 0.0000
Using the z-table, we find the probability associated with the lower bound z-score is approximately 0.4861 and the probability associated with the upper bound z-score is approximately 0.5000.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 with n = 17 is approximately 0.5000 - 0.4861 = 0.0139.
d. To calculate the probability that the sample proportion is between 0.25 and 0.27 with n = 65, we can repeat the same steps as in part c.
Lower Bound z-score = (0.25 - 0.27) / 0.0480 ≈ -0.4167
Upper Bound z-score = (0.27 - 0.27) / 0.0480 = 0.0000
Using the z-table, we find the probability associated with the lower bound z-score is approximately 0.3372 and the probability associated with the upper bound z-score is approximately 0.5000.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 with n = 65 is approximately 0.5000 - 0.3372 = 0.1628.
Consider a population proportion p = 0.27. [You may find it useful to reference the z table.]
a. What is the expected value and the standard error of the sampling distribution of the sample proportion with n = 17 and n = 65? (Round the standard error to 4 decimal places.)
b. Can you conclude that the sampling distribution of the sample proportion is normally distributed for both sample sizes?
multiple choice
Yes, the sampling distribution of the sample proportion is normally distributed for both sample sizes.
No, the sampling distribution of the sample proportion is not normally distributed for either sample size.
No, only the sample proportion with n = 17 will have a normal distribution.
No, only the sample proportion with n = 65 will have a normal distribution.
c. If the sampling distribution of the sample proportion is normally distributed with n = 17, then calculate the probability that the sample proportion is between 0.25 and 0.27. (If appropriate, round final answer to 4 decimal places.)
d. If the sampling distribution of the sample proportion is normally distributed with n = 65, then calculate the probability that the sample proportion is between 0.25 and 0.27. (If appropriate, round final answer to 4 decimal places.)
1 answer
1 answer