The number of diners at a restaurant each day is recorded and a daily average is calculated every month (assume 30 days in a month). the number of diners each day has a mean of 109 and a standard deviation of 53, but does not necessarily follow a normal distribution. the probability that a daily average over a given month is greater than x is 2.5%. calculate x. give your answer to 3 decimal places.

Z scores (converted value in standard normal distribution) can be mapped to probabilities by z tables. The value of x is 212.88 approx.

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 is following normal distribution with mean and standard deviation )

then it can be converted to standard normal distribution as

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

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

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:

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:

The given probability statement is expressed as:

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,

Thus,

The value of x is 212.88 approx.

Learn more about standard normal distributions here:

link

 212.88

Step-by-step explanation:

Given : The probability that a daily average over a given month is greater than x =

The probability that corresponds to  0.025 from a Normal distribution table is 1.96.

Mean :

Standard deviation :

The formula for z-score : -

step-by-step explanation:

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step-by-step explanation: