Question

Assume the random variable x is normally distributed with mean mu equals 50μ=50 and standard deviation sigma equals 7σ=7. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded. upper p left parenthesis upper x greater than 41 right parenthesisp(x>41) which of the follow

Answer 1

Answer:

P(X>41) = 0.9015 or 90.15%

Step-by-step explanation:

We have a mean $\\ \mu =50$ and a standard deviation $\\ \sigma =7$.

To answer the probability P(X>41), or the probability that in a Normal Distribution, with mean $\\ \mu =50$ and standard deviation $\\ \sigma =7$, we have values greater than 41.

We use here values for the standardized normal curve, for which, we know that the z variable $\\ z = \frac{x-\mu}{\sigma}$ describes a Normal Distribution with mean $\\ \mu =0$ and standard deviation $\\ \sigma = 1$.

Having $\\ z = \frac{41-50}{7} = \frac{-9}{7} = -1.2857$

The value z = 1.2857 corresponds to a cumulative probability of 0.4015 (approximately). But we are interested in z = -1.2857.

Since the Normal Distribution is symmetrical, the z-score of 0.4015 indicates the distance from the population mean (the value is 0.4015 units from the mean).

Then, we have to subtract that value from the mean for the z-score value z = -1.2857 (the value is below the mean, denoted by the minus sign). So, the cumulative probability for the latter z-score is 0.5000-0.4015=0.0985. This is the cumulative probability for P(x<41).

Well, the cumulative probability for P(x>41) is 1 - 0.0985 = 0.9015 or 90.15%. The graph below represents this cumulative probability (dark violet area).

Answer 2

The probability of X > 41 for the case when X is normally distributed with mean 50 and standard deviation 7 is 0.9015 approximately.

How can we get probability of a range of a normal distribution?

We can either use the probability function for normal distribution, or we can convert the normal distribution to standard normal distribution, and then use the z-score and the z-table to find the corresponding p-values.

How to get the z scores?

If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.

If we have

$X \sim N(\mu, \sigma)$

(X is following normal distribution with mean $\mu$ and standard deviation $\sigma$)

then it can be converted to standard normal distribution as

$Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)$

(Know the fact that in continuous distribution, probability of a single point is 0, so we can write

$P(Z \leq z) = P(Z < z) )$

Also, know that if we look for Z = z in z tables, the p value we get is

$P(Z \leq z) = \rm p \: value$

For this case, we're specified that:

$X \sim N(\mu = 50, \sigma = 7)$

The probability we need is $P(X > 41)$

We can rewrite it as: $P(X > 41) = 1 - P(X \leq 41)$
Converting X  to Z (standard normal variate), we get:

$Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 50}{7}$

and the needed probability is then obtained as:

$P(X > 41) = 1 - P(X \leq 41) = 1 - P\left( Z \leq \dfrac{41 - 50}{7} \right)\\\\P(X > 41) \approx 1 - P(Z \leq -1.29)$

Using the z-table, we get the p-value for Z = -1.29 as 0.0985

Thus, we get:

$P(X > 41) \approx 1 - P(Z \leq -1.29) = 1- 0.0985 = 0.9015$

Thus, the probability of X > 41 for the case when X is normally distributed with mean 50 and standard deviation 7 is 0.9015 approximately.

Learn more about z-score here:

https://brainly.com/question/21262765

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