Math

QuestionFind the expected number of times you wait over 44 minutes in 29 visits, given mean 42 min and SD 5 min.

Studdy Solution

STEP 1

Assumptions1. The waiting time for food is normally distributed. The mean waiting time is42 minutes3. The standard deviation is5 minutes4. You visit the restaurant29 times5. We are interested in the number of times you would wait longer than44 minutes

STEP 2

First, we need to calculate the z-score for the waiting time of44 minutes. The z-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formulaZ=XμσZ = \frac{X - \mu}{\sigma}where X is the value we are interested in, μ is the mean, and σ is the standard deviation.

STEP 3

Now, plug in the given values for X, μ, and σ to calculate the z-score.
Z=44425Z = \frac{44 -42}{5}

STEP 4

Calculate the z-score.
Z=4442=0.4Z = \frac{44 -42}{} =0.4

STEP 5

The z-score tells us that44 minutes is0.4 standard deviations above the mean. We need to find the probability of waiting more than44 minutes. This is equivalent to finding the area under the normal curve to the right of the z-score of0.4. We can find this by subtracting the cumulative probability of the z-score from1.
(X>44)=1(Z<0.4)(X >44) =1 -(Z <0.4)

STEP 6

We can use a standard normal distribution table or a calculator to find the cumulative probability of the z-score0.4. Let's assume it is0.6554.
(X>44)=10.6554(X >44) =1 -0.6554

STEP 7

Calculate the probability of waiting more than44 minutes.
(X>44)=10.6554=0.3446(X >44) =1 -0.6554 =0.3446

STEP 8

Now that we have the probability of waiting more than44 minutes, we can find the expected number of times you would wait more than44 minutes by multiplying this probability by the total number of visits (29).
Expectednumberoftimes=(X>44)×NumberofvisitsExpected\, number\, of\, times =(X >44) \times Number\, of\, visits

STEP 9

Plug in the values for the probability and the number of visits to calculate the expected number of times.
Expectednumberoftimes=.3446×29Expected\, number\, of\, times =.3446 \times29

STEP 10

Calculate the expected number of times you would wait more than44 minutes.
Expectednumberoftimes=0.3446×29=9.9934Expected\, number\, of\, times =0.3446 \times29 =9.9934

STEP 11

Since we can't have a fraction of a visit, we round this to the nearest whole number.
Expectednumberoftimes=10Expected\, number\, of\, times =10You would expect to wait longer than44 minutes10 times.

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