What is wrong with the following statements?
a) For a large n, the distribution of observed values will be approximately Normal
b) 68-95-99.7 rule says that "x bar" should be within m (mean) +/- 2 SD (standard deviation) about 95% of the time
c) The central limit theorem states that for large n, m is approximately Normal.
a) This one should state that the mean of the observed values will be approximately Normal, not that the distribution will be approximately Normal.
b) I don't know what the 68-95-99.7 rule actually is, but it'll probably say something about 68%, 95% and 99.7% of the distribution lying within one, two and three standard deviations of the mean respectively. It certainly won't say that the sample mean should lie within +/- 2 standard deviations of the mean of the actual mean about 95% of the time (even though it's actually true). It sounds to me as though the wrong rule is being quoted here.
c) The central limit theorem states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. I'll guess that the problem with the statement given is that it doesn't say anything about how large is "large" needs to be, and it doesn't say anything about m being the mean of a number of independent random variables with finite mean and variance.
a) The statement is incorrect because the distribution of observed values is not necessarily normal, even for a large sample size. It depends on the underlying population distribution.
b) There are two issues with this statement. Firstly, "x bar" should be within m +/- 2 SD about 95% of the time is misleading. The correct statement should say that approximately 95% of the sample means (x bar) will fall within m +/- 2 SD. Secondly, the 68-95-99.7 rule actually refers to the percentage of values within 1, 2, and 3 standard deviations from the mean in a normal distribution, not specifically to sample means.
c) The statement is incorrect. The central limit theorem states that for large n, the sampling distribution of the sample mean (x bar) will be approximately normal, regardless of the shape of the underlying population distribution. It does not make any reference to the population mean (m) being normal.
a) The statement "For a large n, the distribution of observed values will be approximately Normal" is incorrect because it is a misunderstanding of the Central Limit Theorem (CLT). The CLT actually states that if you have a large enough sample size (n) and the observations are independent and identically distributed, then the distribution of the sample mean will approach a Normal distribution.
b) The 68-95-99.7 rule, also known as the empirical rule, is commonly used to describe the percentage of data within certain standard deviations from the mean in a Normal distribution. However, the statement "68-95-99.7 rule says that 'x bar' should be within m (mean) +/- 2 SD (standard deviation) about 95% of the time" is incorrect. The rule applies to individual observations, not the sample mean.
c) The statement "The central limit theorem states that for large n, m is approximately Normal" is incorrect. The central limit theorem actually states that for large sample sizes, the sampling distribution of the sample mean will be approximately Normal, regardless of the shape of the population distribution. It does not make any specific claims about the population mean being normally distributed.
a) The statement is wrong because the distribution of observed values is not necessarily always approximately normal for large n. The normal approximation relies on certain conditions being met, such as the observations being independent and identically distributed.
To determine if the distribution of observed values will be approximately normal, you can conduct a visual inspection by creating a histogram or a Q-Q plot to check for symmetry and the shape of the distribution. Alternatively, you can use statistical tests like the Shapiro-Wilk test or the Anderson-Darling test to assess normality.
b) The statement is wrong. According to the 68-95-99.7 rule, approximately 68% of the values should fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and around 99.7% within 3 standard deviations. It is not specific to the sample mean, denoted as "x bar." The rule applies to any normally distributed random variable.
To determine if a particular value falls within these ranges, you need to calculate the mean and standard deviation of the distribution and then use these values to compute the intervals.
c) The statement is incorrect. The Central Limit Theorem (CLT) states that for a large enough sample size n, the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution. It does not say that the mean itself is approximately normal for large n.
To apply the CLT, you need to ensure that the sample size is sufficiently large (typically n ≥ 30), and the observations are independent or come from a population with a finite variance. Then, you can use the sample mean to approximate the population mean with a certain level of confidence, assuming that the required conditions are met.