Let x represent the number which shows up when a balanced die is rolled. Then x is a random variable with a uniform distribution. Let x-bar denote the mean of the numbers obtained when the die is rolled 32 times. For samples of size 32, which of the following statements concerning the sampling distribution of the mean is true?
A) x-bar is approximately normally distributed.
B) The distribution of x-bar is uniform.
C) x-bar is normally distributed.
D) None of the above statements is true.
I am thinking it is uniformed because of the sample size of 32 which means if this was to be on a dotplot, there would only be one plotted data because the dice was rolled 32 times.
No. X-bar is the running mean. Each time a number is thrown, it changes the mean. It should approximate a normal distribution.
ANSWER: x̅ is approximately normally distributed.
You are correct. The statement "A) x-bar is approximately normally distributed" is true. According to the Central Limit Theorem, for a large enough sample size, the sampling distribution of the mean will approach a normal distribution, regardless of the shape of the population distribution. Since the sample size in this case is 32, which is considered large, the sampling distribution of x-bar will be approximately normally distributed. Therefore, option A is the correct choice.
To determine which statement is true, we need to understand the concept of the sampling distribution of the mean.
The sampling distribution of the mean refers to the distribution of the means calculated from multiple samples taken from the same population. In this case, we are rolling a balanced die 32 times and calculating the mean (x-bar) of the numbers obtained.
The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the mean will be approximately normally distributed, regardless of the shape of the population distribution. This is true regardless of whether the individual measurements are themselves normally distributed or not.
The key here is that the sample size is 32, which is sufficiently large. Therefore, the correct answer is A) x-bar is approximately normally distributed.
While the distribution of the individual measurements (x) is uniform, the sampling distribution of the mean (x-bar) will tend to approximate a normal distribution. The more samples we take, the closer the sampling distribution of the mean will resemble a normal distribution. So, option B) "The distribution of x-bar is uniform" is not correct.
Hence, the correct answer is A) x-bar is approximately normally distributed.