Respuesta :
Answer: 212.88
Step-by-step explanation:
Given : The probability that a daily average over a given month is greater than x = [tex]2.5\%=0.025[/tex]
The probability that corresponds to 0.025 from a Normal distribution table is 1.96.
Mean : [tex]\mu = 109[/tex]
Standard deviation : [tex]\sigma = 53[/tex]
The formula for z-score : -[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
[tex]\Rightarrow\ 1.96=\dfrac{x-109}{53}\\\\\Rightarrow\ x=53\times1.96+109\\\\\Rightarrow\ x=212.88[/tex]
Z scores (converted value in standard normal distribution) can be mapped to probabilities by z tables. The value of x is 212.88 approx.
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
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
For the given case, let the random variable X tracks the number of dinners at given restaurant. Assuming normal distribution being pertained by X, we get:
[tex]X \sim N(109, 53)[/tex]
The given data shows that:
2.5% of all daily averages records lie bigger than value X = x
or
P(X > x) = 2.5% 0.025
Converting it to standard normal distribution(so that we can use z tables and p values to get the unknown x), we get:
[tex]z = \dfrac{x-\mu}{\sigma} = \dfrac{x - 109}{53}[/tex]
The given probability statement is expressed as:
[tex]P(Z > z) = 2.5\% = 0.025\\P(Z \leq z) = 1 - 0.025 = 0.975[/tex]
Seeing the z tables, we will try to find at what value of z, the p value is obtained near to 0.975
We get z = 1.96.
Thus,
[tex]z = 1.96 = \dfrac{x - 109}{53}\\\\x = 1.96 \times 53} + 109 = 212.88[/tex]
Thus,
The value of x is 212.88 approx.
Learn more about standard normal distributions here:
https://brainly.com/question/10984889
