Let the random variable X follow a normal distribution with ?= 80 and ?2= 100.

[statpr226]

Please help with a solution! No way a proper answer is coming out, I'm preparing for exams!
1)Let the random variable X follow a normal distribution with ?= 80 and ?2= 100. a.Find the probability that X is greater than 60. Draw a probability distribution graph.b. Find the probability that X is greater than 72 and less than 82.c. Find the probability that X is less than 55.d. The probability is 0.1 that X is greater than what number? e. The probability is 0.6826 that X is in the symmetric interval about the mean between which two numbers?

2) Candidates for employment at a city fire department are required to take a written aptitude test. Scores on this test are normally distributed with a mean of 280 and a standard deviation of 60. A random sample of nine test scores was taken. •a. What is the standard error of the sample mean score?•b. What is the probability that the sample mean score is less than 270?•c. What is the probability that the sample mean score is more than 250?•d. Suppose that the population standard deviation is, in fact, 40, rather than 60. Without doing the calculations, state how this would change your answers to parts (a), (b), and (c). Illustrate your conclusions with the appropriate graphs.

 

[HallsofIvy]

For a normal distribution with mean 80 and standard deviation 10, use a "standard normal distribution (mean 0 and standard deviation 1), with \(\displaystyle \frac{x- 80}{10}\) instead of x. There is a nice table of the standard normal distribution, together with an interactive graph, at https://www.mathsisfun.com/data/standard-normal-distribution-table.html

With x= 82, \(\displaystyle \frac{x- 80}{10}= \frac{2}{10}= 0,2\). Using the interactive graph above I get probability 0.0793. With x= 72, \(\displaystyle \frac{x- 80}{10}= \frac{-8}{10}= -0,8\). Using the interactive graph above I get probability -02881.

The probability the value lies between 72 and 82 is 0.0793- (-0.2881)= 0.0793+ 0.2881= 0.3674.

Now, try the second problem.