An SRS of size n is taken from a large population whose distribution of income is extremely right-skewed and the mean income is calculated. Which of the following statements is false?
a) When n > 30, the sampling distribution of is approximately normal.
b) When n increases, the sample standard deviation decreases (s).
c) The standard deviation of the sampling distribution of is .
d) The standard error is the standard deviation of the sample (s).
e) When n increases, the standard deviation of the sampling distribution of decreases.
Well, that's a lot of statistical jargon! Let's take it one step at a time, shall we?
a) When n > 30, the sampling distribution of [mean income] is approximately normal.
True! This statement is actually true. When the sample size is greater than 30, the Central Limit Theorem kicks in and the sampling distribution of the mean becomes approximately normal.
b) When n increases, the sample standard deviation decreases (s).
Hmm, false! The sample standard deviation (s) actually has nothing to do with the sample size (n). Increasing the sample size does not necessarily mean the sample standard deviation will decrease.
c) The standard deviation of the sampling distribution of [mean income] is [???].
Unclear! You didn't provide the value for the standard deviation, so I can't say if it's true or false.
d) The standard error is the standard deviation of the sample (s).
False! The standard error is actually the standard deviation of the sampling distribution of the mean, not the standard deviation of the sample itself.
e) When n increases, the standard deviation of the sampling distribution of [mean income] decreases.
True! As the sample size increases, the standard deviation of the sampling distribution of the mean decreases. So, this statement is true.
I hope that helps clarify things a bit!
The false statement is c) The standard deviation of the sampling distribution of is sqrt(n).
Explanation:
a) When n > 30, the sampling distribution of is approximately normal. This statement is true because of the Central Limit Theorem, which states that when the sample size is large enough, the sampling distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution.
b) When n increases, the sample standard deviation decreases (s). This statement is generally true because as the sample size increases, there is more information available, which tends to reduce the variability in the sample.
c) The standard deviation of the sampling distribution of is sqrt(n). This statement is false. The standard deviation of the sampling distribution of the sample mean is actually equal to the population standard deviation divided by the square root of the sample size (standard deviation / sqrt(n)).
d) The standard error is the standard deviation of the sample (s). This statement is false. The standard error is actually equal to the standard deviation of the sampling distribution of the sample mean and is given by the formula standard deviation / sqrt(n), where n is the sample size.
e) When n increases, the standard deviation of the sampling distribution of decreases. This statement is generally true because as the sample size increases, the standard deviation of the sampling distribution of the sample mean tends to decrease, indicating that the sample mean is becoming a more precise estimate of the population mean.
To determine which statement is false, let's examine each option:
a) When n > 30, the sampling distribution of is approximately normal.
This statement is true. According to the Central Limit Theorem, when the sample size (n) is large (typically greater than 30), the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution.
b) When n increases, the sample standard deviation decreases (s).
This statement is not necessarily true. The sample standard deviation (s) measures the spread of values within a sample. Increasing the sample size (n) does not guarantee a decrease in sample standard deviation (s). The sample standard deviation depends on the variability of the data, not solely on the sample size.
c) The standard deviation of the sampling distribution of .
This statement is true. The standard deviation of the sampling distribution of the mean, denoted as σ/√n, where σ is the population standard deviation, measures the spread of the means from multiple samples in relation to the true population mean.
d) The standard error is the sample standard deviation (s).
This statement is false. The standard error (SE) is not the same as the sample standard deviation (s). The standard error is calculated as the standard deviation of the sampling distribution of the mean, which is σ/√n.
e) When n increases, the standard deviation of the sampling distribution of decreases.
This statement is true. As the sample size (n) increases, the standard deviation of the sampling distribution of the mean decreases. This indicates that larger sample sizes lead to more accurate estimates of the population mean.
Based on the above explanations, the false statement is d) The standard error is the standard deviation of the sample (s).
Typos?
C is incomplete.
I assume that (s) indicates your answer.
Agree with d and b, but also would include a.
How is e different than b?