Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 7.3 The Sampling Distribution of the Sample Mean X This section focuses on when a variable under consideration is normally distributed or is not normally distributed and how that affects the sampling distribution. Key Fact 7.2 Sampling Distribution of the Sample Mean for a Normally Distributed Variable Suppose that a variable X of a population is normally distributed with mean . Then for samples of size n the variable X is also normally distributed and has mean and standard deviation and standard deviation n Example The distribution for a 10 kilometer run in New York City is normally distributed. The mean time to finish the race is standard deviation is 61 minutes and the 9 minutes . Find the following: a. The population distribution b. The sampling distribution of the sample mean for samples of size 4 c. The sampling distribution of the sample mean for samples of size 9 d. What do you notice about the population distribution and the two sampling distributions? We have been focusing on normal distributions and how changing the sample size for a sampling distribution can affect the shape. The next question to ask is how a non-normal distribution is affected by sampling distributions of various sizes Why don’t we look at sampling distributions of size 10, 20 and 30 if the variable under consideration’s shape is uniform or is j-shaped. These results bring us to the Center Limit Theorem Key Fact 7.3 Central Limit Theorem For a relatively large sample size the variable X is approximately normally distributed, regardless of the distribution of the variable under consideration. This approximation becomes better with increasing sample size. Example 1 a. For 10 kilometer New York City race, what is the probability that the sampling error made in estimation the population mean finishing time by that of a random sample of four runners will be no more than five minutes off from the true population mean time. The mean time to finish the race is 61 minutes and the standard deviation is 9 minutes . b. If the random sample is changed from four runners to nine runners, what is the probability that the sampling error made in estimation the population mean finishing time by that of a random sample of nine runners will be no more than five minutes off from the true population mean time. Example 2 Key Fact 7.4 Sampling Distribution of the Sample Mean Suppose that a variable X of a population has mean deviation . Then, for samples of size n The mean of and standard X equals the population mean, or X The standard deviation of X equals the population standard deviation divided by the square root of the sample size, or X If n X is normally distributed, so is X , regardless of sample size; and If the sample size is large X is approximately normally distributed, regardless of the distribution of X