# 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. (e) The sampling distribution shows how the sample was distributed around the sample mean.

Question
Sampling distributions
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.
(e) The sampling distribution shows how the sample was distributed around the sample mean.

2021-03-05
(a) Correct, because a larger sample gives more information about the population and thus allows us to make more accurate predictions. If the predictions are more accurate, this also means that the variability (measured by the standard deviation) is also decreased.
(b) Correct, the standard deviation of the sampling distribution of the sample mean is the variability of the sample mean over all possible samples.
(c) Correct, the sample mean is an unbiased estimator of the population mean.
(d) Correct, because the the variation is given by the standard deviation of the sampling distribution.
(e) Incorrect, because the sampling distribution is the distribution of all possible sample means, which cannot be centered about the sample mean (since there are many sample means).

### Relevant Questions

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 distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85.
Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?
Both distributions are approximately normal with mean 65 and standard deviation 3.5.
A
Both distributions are approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
B
Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.
C
Only the distribution for size 85 is approximately normal. Both distributions have mean 65 and standard deviation 3.5.
D
Only the distribution for size 85 is approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
E
Which of the following are correct general statements about the Central Limit Theorem?
(Select all that apply. To be marked correct: All of the correct selections must be made, with no incorrect selections.)
Question 3 options:
Its name is often abbreviated by the three capital letters CLT.
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.
It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
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.
Which of the following are correct general statements about the central limit theorem? Select all that apply
1. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$
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.
Which of the following are correct general statements about the Central Limit Theorem? Select all that apply.
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.
Which of the following are correct general statements about the central limit theorem? Select all that apply
1. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$
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.
Which of the following is true about the sampling distribution of means?
A. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is.
B. Sampling distributions of means are always nearly normal.
C. Sampling distributions of means get closer to normality as the sample size increases.
D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.
Give full and correct answer for this questions 1) A t-test is a ? 2) Which of the following statement is true? a)The less likely one is to commit a type I error, the more likely one is to commit a type II error, b) A type I error has occurred when a false null hypothesis has been wrongly accepted. c) A type I error has occurred when a two-tailed test has been performed instead of a one-tailed test, d) None of the above statements is true. 3)Regarding the Central Limit Theorem, which of the following statement is NOT true? a.The mean of the population of sample means taken from a population is equal to the mean of the original population. b. The frequency distribution of the population of sample means taken from a population that is not normally distributed will approach normality as the sample size increases. c. The standard deviation of the population of sample means is equal to the standard deviation of the, original population. d. The frequency distribution of the population of sample means taken from a population that is not normally distributed will show less dispersion as the sample size increases.
Indicate true or false for the following statements. If false, specify what change will make the statement true.
a) In the two-sample t test, the number of degrees of freedom for the test statistic increases as sample sizes increase.
b) When the means of two independent samples are used to to compare two population means, we are dealing with dependent (paired) samples.
c) The $$\displaystyle{x}^{{{2}}}$$ distribution is used for making inferences about two population variances.
d) The standard normal (z) score may be used for inferences concerning population proportions.
e) The F distribution is symmetric and has a mean of 0.
f) The pooled variance estimate is used when comparing means of two populations using independent samples.
g) It is not necessary to have equal sample sizes for the paired t test.