The Normal distribution: Practice Questions

The continuous random variable \(X\) is normally distributed with mean \(20\) and standard deviation \(4\).
Find \(P(X > 26)\).

The random variable \(X\) follows a normal distribution \(X \sim N(\mu, \sigma^2)\). Given that \(P(X < 30) = 0.1587\) and \(P(X > 70) = 0.1587\), find the value of \(\mu\) and \(\sigma\).

The continuous random variable \(X\) is normally distributed with mean 100 and standard deviation 15. Calculate the conditional probability \(P(X > 115 | X > 100)\), giving your answer to 3 significant figures.

The continuous random variable \(X\) is normally distributed with mean \(50\) and variance \(100\).
Find the standardized value (z-score) corresponding to \(X = 65\).

The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(P(X < 40) = 0.0668\) and \(P(X > 60) = 0.3085\), find the value of \(\mu\). (Assume the corresponding standard normal \(z\)-scores are \(P(Z < -1.5) = 0.0668\) and \(P(Z < 0.5) = 0.6915\)).

The random variable \(X\) follows a normal distribution such that \(X \sim N(\mu, \sigma^2)\). Given that \(P(X < 45) = 0.7\) and \(\mu = 40\), find the value of the standard deviation \(\sigma\) correct to 2 decimal places.

A IQ test is designed so that the scores are normally distributed with a mean of 100 and a standard deviation of 15.
(a) Find the probability that a person has an IQ between 90 and 120.
(b) A group of 500 people take the test. Estimate how many people would be expected to have an IQ over 130.