Question

Suppose s leftskewed distribution exists with a mean of 133 and standard deviation of 19. Consider a random sample of 96 observations. Use this irformation to determine the following probabilities. Aound solutions to four decimal pleces, if necessary. Find the probabitity that a random sample of sise $ hat a mean laws than 123. P(x<12δ) Find the probability that a random sample of aite i4 has a man between 133 and 138 . P(133

Answer 1

To find the probability of a random sample having a mean less than 123 in a left-skewed distribution with given parameters, we can use the Central Limit Theorem and z-scores.

Step-by-step explanation:

To determine the probability that a random sample of size $ hat a mean less than 123 in a left-skewed distribution with a mean of 133 and standard deviation of 19, we can use the Central Limit Theorem. The Central Limit Theorem states that as sample size increases, the distribution of sample means approaches a normal distribution.

First, we need to find the z-score corresponding to a sample mean of 123. Using the formula z = (x - mu) / (sigma / sqrt(n)), where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size, we get:

z = (123 - 133) / (19 / sqrt(96))

After calculating the z-score, we can use a standard normal distribution table or software to find the probability corresponding to a z-score less than the calculated z-score. This will give us the probability that a random sample of size 96 has a mean less than 123.