The correct statement which is incorrect from the options about the sampling distribution of the sample mean (a) the standard deviation of the sampling distribution will decrease as the sample size increases, (b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples, (c) the sample mean is an unbiased estimator of the true population mean, (d) the sampling distribution shows how the sample mean will vary in repeated samples, (e) the sampling distributions shows how the sample was distributed around the sample mean.

The correct statement which is incorrect from the options about the sampling distribution of the sample mean (a) the standard deviation of the sampling distribution will decrease as the sample size increases, (b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples, (c) the sample mean is an unbiased estimator of the true population mean, (d) the sampling distribution shows how the sample mean will vary in repeated samples, (e) the sampling distributions shows how the sample was distributed around the sample mean.

Question
Sampling distributions
asked 2021-03-09
The correct statement which is incorrect from the options about the sampling distribution of the sample mean
(a) the standard deviation of the sampling distribution will decrease as the sample size increases,
(b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples,
(c) the sample mean is an unbiased estimator of the true population mean,
(d) the sampling distribution shows how the sample mean will vary in repeated samples,
(e) the sampling distributions shows how the sample was distributed around the sample mean.

Answers (1)

2021-03-10
Here, for option (a):
This option is correct because if a sample is larger, it provides more information about the population which allows us to predict more accurate values. This means that the variability decreases if the predictions are more accurate.
For option (b):
This option is correct because the standard deviation of a sampling distribution of a sample mean and the variability of the sample mean over all possible samples are same.
For option (c):
This option is correct because a population mean has an unbiased estimator which is the sample mean.
For option (d):
This option is correct because the variation is given by the standard deviation of the sampling distribution.
For option (e):
This option is incorrect because the distribution of all possible sample mean is the sampling distribution and it cannot be centered about the sample mean (since there are many sample means).
Hence, the correct answer is (e).
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Relevant Questions

asked 2021-03-04
Which of the following statements about the sampling distribution of the sample mean is incorrect?
(a) The standard deviation of the sampling distribution will decrease as the sample size increases.
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
(c) The sample mean is an unbiased estimator of the population mean.
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
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Previous studies show that \( \sigma_1 = 19 \).
For Englewood (a suburb of Denver), a random sample of \( n_2 = 12 \) winter days gave a sample mean pollution index of \( x_2 = 37 \).
Previous studies show that \( \sigma_2 = 13 \).
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\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)
\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)
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The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
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At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
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At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are not statistically significant.
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Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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Both distributions are approximately normal with mean 65 and standard deviation 3.5.
A
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B
Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.
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Question 3 options:
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The accuracy of the approximation it provides, improves as the sample size n increases.
The word Central within its name, is meant to signify its role of central importance in the mathematics of probability and statistics.
It is a special example of the particular type of theorems in mathematics, which are called Limit Theorems.
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The accuracy of the approximation it provides, improves when the trial success proportion p is closer to \(50\%\).
It specifies the specific shape of the curve which approximates certain sampling distributions.
It specifies the specific mean of the curve which approximates certain sampling distributions.
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D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.
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2. It specifies the specific mean of the curve which approximates certain sampling distributions.
3. It is a special example of the particular type of theorems in mathematics, which are called Limit theorems.
4. It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
5. It’s name is often abbreviated by the three capital letters CLT.
6. The accuracy of the approximation it provides, improves as the sample size n increases.
7. The word Central within its name, is mean to signify its role of central importance in the mathematics of probability and statistics.
8. It specifies the specific shape of the curve which approximates certain sampling distributions.
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1. It specifies the specific shape of the curve which approximates certain sampling distributions.
2. It’s name is often abbreviated by the three capital letters CLT
3. The word Central within its name, is meant to signify its role of central importance in the mathematics of probability and statistics.
4. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to 50\%.
5. It specifies the specific mean of the curve which approximates certain sampling distributions.
6. The accuracy of the approximation it provides, improves as the sample size n increases.
7. It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
8. It is a special example of the particular type of theorems in mathematics, which are called limit theorems.
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-Sampling distributions are always nearly normal.
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-Sampling distributions get closer to normality as the sample size increases.
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