STATISTICS: I'm struggling with how to calculate the following problems:
*Assume that X~N(3,1.5)
For what value of X is Pr(x<K) = 50%
*Assume that X~N(3,1.5)
What value of X would be equal to a z-score of -2?
*Assume that X~N(3,1.5)
Find the 90% percentile
*Assume X~N(4.5,3) Assume you take a sample of size 20; what is the expected distribution?
To solve these problems, we need to understand the concept of the normal distribution and how it is related to probability.
1. For what value of X is Pr(x < K) = 50%?
To find the value of X for which the probability of X being less than K is 50%, we will use the standard normal distribution table or a statistical software. First, we need to standardize the value of K into a z-score using the formula: z = (K - mean) / standard deviation. In this case, the mean is 3 and the standard deviation is 1.5. Then, we check the standard normal distribution table for the z-score that corresponds to a probability of 0.50 (since Pr(x < K) = 50%). Finally, we multiply the z-score by the standard deviation and add the mean to get the value of X.
2. What value of X would be equal to a z-score of -2?
To find the value of X corresponding to a specific z-score, we use the formula: X = z * standard deviation + mean. In this case, the z-score is -2, and the mean and standard deviation are given as 3 and 1.5, respectively. Substituting these values into the formula will give us the value of X.
3. Find the 90th percentile.
To find the 90th percentile, we need to determine the value of X for which 90% of the values fall below it. We can use the standard normal distribution table to find the corresponding z-score. From the standard normal distribution table, we look for the z-score that corresponds to a cumulative probability of 0.90. Once we have the z-score, we use the formula X = z * standard deviation + mean to find the value of X.
4. What is the expected distribution if we take a sample of size 20 from X ~ N(4.5, 3)?
When you take a sample from a normal distribution, the distribution of the sample means will also follow a normal distribution. This is called the sampling distribution of the mean. The mean of the sampling distribution will be the same as the mean of the original distribution, and the standard deviation of the sampling distribution (also known as the standard error) will be the standard deviation of the original distribution divided by the square root of the sample size. In this case, the expected distribution of the sample means will be a normal distribution with a mean of 4.5 and a standard deviation of 3 divided by the square root of 20.