Question

If X represents a random variable coming from a normal distribution and P(X<7.8)= 0.55, then P(X>7.8)= 0.45. True or False

Answer 1

Explanation:

The statement is true. Here's an explanation:

If X represents a random variable coming from a normal distribution, we can use the properties of the standard normal distribution to find probabilities. The standard normal distribution has a mean of 0 and a standard deviation of 1.

We can convert any normal distribution to the standard normal distribution using the formula:

z = (x - μ) / σ

where x is a value from the normal distribution, μ is the mean of the normal distribution, σ is the standard deviation of the normal distribution, and z is the corresponding value on the standard normal distribution.

In this case, we know that P(X < 7.8) = 0.55. We can convert this to the standard normal distribution by subtracting the mean and dividing by the standard deviation:

P((X - μ) / σ < (7.8 - μ) / σ) = 0.55

We can use a standard normal distribution table (or calculator) to find the corresponding z-score for the probability 0.55. The closest value we can find in the table is 0.54, which corresponds to a z-score of 0.13.

So we have:

P(Z < 0.13) = 0.55

Now we can use the properties of the standard normal distribution to find P(Z > 0.13). Since the total area under the standard normal distribution curve is 1, we know that:

P(Z < 0.13) + P(Z > 0.13) = 1

Therefore:

P(Z > 0.13) = 1 - P(Z < 0.13) = 1 - 0.55 = 0.45

Since X and Z are related by the formula z = (x - μ) / σ, we can conclude that:

P(X > 7.8) = P((X - μ) / σ > (7.8 - μ) / σ) = P(Z > 0.13) = 0.45

Therefore, the statement "P(X > 7.8) = 0.45" is true.

Answer 2

The statement is true. If P(X < 7.8) = 0.55 in a normal distribution, then P(X > 7.8) is the complement, equal to 1 - 0.55 = 0.45, ensuring the total probability remains 1.

False. In a normal distribution, the total area under the curve is 1. If P(X < 7.8) = 0.55, it means the probability that the random variable X is less than 7.8 is 0.55. Therefore, P(X > 7.8) would be the remaining probability under the curve, which is 1 - 0.55 = 0.45.

So, P(X > 7.8) is indeed 0.45, consistent with the complement rule in probability. The statement is true, not false.

It's important to note that the probabilities are complementary; if the probability of being less than 7.8 is 0.55, then the probability of being greater than 7.8 is 0.45, and vice versa. The sum of these probabilities is always 1 in a normal distribution.