Consider a random sample with sample mean x. If the sample size increased from n = 40 to n = 360, then the standard deviation of x will:
a. remain the same
b. increase by a factor of 9 (will be multiplied by 9)
c. decrease by a factor of 9 (will be multiplied by 1/9)
d. decrease by a factor of 3 (will be multiplied by 1/3)
Assuming that you are referring to the standard deviation and not the standard error of the mean, the standard deviation will remain essentially the same.
To determine how the standard deviation of x changes when the sample size increases, we need to understand the relationship between the standard deviation and sample size.
The standard deviation of a sample mean, often denoted as s_x̄, is given by the formula σ/√n, where σ represents the population standard deviation and n is the sample size.
In this case, the sample size increases from n = 40 to n = 360. Since the sample size is in the denominator of the formula, as n increases, the standard deviation of x, s_x̄, will decrease.
To determine the factor by which the standard deviation changes, we can calculate the ratio of the standard deviations:
Ratio = (s_x̄ with n = 360) / (s_x̄ with n = 40)
Since the ratio represents how much the standard deviation changes, we can compare it to the given options to determine the correct answer.
Let's perform the calculations:
If the ratio is greater than 1, it means the standard deviation increased.
If the ratio is less than 1, it means the standard deviation decreased.
Ratio = (s_x̄ with n = 360) / (s_x̄ with n = 40) = √(360/40) = √9 = 3
From the calculation, we find that the ratio is equal to 3. Therefore, the correct answer is:
d. decrease by a factor of 3 (will be multiplied by 1/3)