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.
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The false statement is d) The standard error is the standard deviation of the sample (s).
Explanation:
a) This statement is true. According to the Central Limit Theorem, when the sample size is large (typically considered greater than 30), the sampling distribution of the sample mean approximates a normal distribution, regardless of the shape of the population distribution.
b) This statement is true. As the sample size increases, the variability in the sample decreases, resulting in a smaller sample standard deviation (s). This is because larger sample sizes provide more information and tend to better represent the population.
c) This statement is true. The standard deviation of the sampling distribution of the sample mean, denoted as σ/√n (where σ is the population standard deviation and n is the sample size), represents the variability of sample means that would be obtained if multiple samples of size n were taken from the population.
d) This statement is false. The standard error represents the standard deviation of the sampling distribution of the sample mean, not the standard deviation of the sample itself (s). The standard error is calculated as σ/√n, where σ is the population standard deviation and n is the sample size.
e) This statement is true. As the sample size increases, the standard deviation of the sampling distribution of the sample mean decreases. This is because larger sample sizes provide more information and reduce the variability of sample means, resulting in a more precise estimate of the population mean.
To determine which statement is false, let's evaluate each statement one by one:
a) When n > 30, the sampling distribution of is approximately normal.
This statement is true. When the sample size (n) is larger than 30, the sampling distribution of the mean becomes approximately normally distributed, regardless of the shape of the population distribution. This property is known as the Central Limit Theorem.
b) When n increases, the sample standard deviation decreases (s).
This statement is false. Increasing the sample size (n) does not necessarily lead to a decrease in the sample standard deviation (s). The sample standard deviation measures the dispersion of the data points within a single sample, and it is not directly affected by the sample size.
c) The standard deviation of the sampling distribution of is .
This statement is true. The standard deviation of the sampling distribution of the mean is often denoted as σ/√n (where σ represents the population standard deviation). It represents the average amount of variation in the means of different samples taken from the population.
d) The standard error is the standard deviation of the sample (s).
This statement is false. The standard error represents the standard deviation of the sampling distribution of the mean, and it is denoted as σ/√n or s/√n. On the other hand, the standard deviation of the sample (s) measures the variability within a single sample.
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 yield estimates of the population mean that are more precise and less variable.