QuestionFind how many days Mamadou's commute is between 23 and 25 minutes in a year, given mean 28 and SD 3.

Studdy Solution

STEP 1

Assumptions1. The time it takes for Mamadou to commute to work is normally distributed. . The mean commute time is28 minutes.
3. The standard deviation of the commute time is3 minutes.
4. Mamadou commutes to work210 days per year.
5. We want to find out how many times his commute would be between23 and25 minutes.

STEP 2

First, we need to standardize the commute times of23 and25 minutes using the Z-score formula. The Z-score is a measure of how many standard deviations an element is from the mean.
$Z = \frac{X - \mu}{\sigma}$ where- Z is the Z-score, - X is the value we want to standardize, - μ is the mean, and- σ is the standard deviation.

STEP 3

Calculate the Z-score for23 minutes.
$Z_{23} = \frac{23 -28}{3}$

STEP 4

Calculate the Z-score for25 minutes.
$Z_{25} = \frac{25 -28}{3}$

STEP 5

Now, we need to find the area between these two Z-scores under the standard normal distribution curve. This area represents the probability that Mamadou's commute time is between23 and25 minutes. We can find this area using a Z-table or a calculator with a normal distribution function.

STEP 6

The area under the curve between $Z_{23}$ and $Z_{25}$ is the difference between the area to the left of $Z_{25}$ and the area to the left of $Z_{23}$ . $(23 < X <25) =(Z_{25}) -(Z_{23})$

STEP 7

To find the number of days Mamadou's commute is between23 and25 minutes, we multiply the total number of commuting days by the probability we just calculated.
$Number\, of\, days = Total\, number\, of\, days \\times(23 < X <25)$

STEP 8

Plug in the values for the total number of days and the probability to calculate the number of days.
$Number\, of\, days =210 \\times [(Z_{25}) -(Z_{23})]$

STEP 9

The result should be rounded to the nearest whole number, as it is not possible to have a fraction of a day.

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QuestionFind how many days Mamadou's commute is between 23 and 25 minutes in a year, given mean 28 and SD 3.

Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the Z-score for23 minutes.
$Z_{23} = \frac{23 -28}{3}$

STEP 4

Calculate the Z-score for25 minutes.
$Z_{25} = \frac{25 -28}{3}$

STEP 5

Now, we need to find the area between these two Z-scores under the standard normal distribution curve. This area represents the probability that Mamadou's commute time is between23 and25 minutes. We can find this area using a Z-table or a calculator with a normal distribution function.

STEP 6

The area under the curve between $Z_{23}$ and $Z_{25}$ is the difference between the area to the left of $Z_{25}$ and the area to the left of $Z_{23}$ . $(23 < X <25) =(Z_{25}) -(Z_{23})$

STEP 7

To find the number of days Mamadou's commute is between23 and25 minutes, we multiply the total number of commuting days by the probability we just calculated.
$Number\, of\, days = Total\, number\, of\, days \\times(23 < X <25)$

STEP 8

Plug in the values for the total number of days and the probability to calculate the number of days.
$Number\, of\, days =210 \\times [(Z_{25}) -(Z_{23})]$

STEP 9

The result should be rounded to the nearest whole number, as it is not possible to have a fraction of a day.

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QuestionFind how many days Mamadou's commute is between 23 and 25 minutes in a year, given mean 28 and SD 3.

Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the Z-score for23 minutes.Z23=23−283Z_{23} = \frac{23 -28}{3}Z23​=323−28​

STEP 4

Calculate the Z-score for25 minutes.Z25=25−283Z_{25} = \frac{25 -28}{3}Z25​=325−28​

STEP 5

Now, we need to find the area between these two Z-scores under the standard normal distribution curve. This area represents the probability that Mamadou's commute time is between23 and25 minutes. We can find this area using a Z-table or a calculator with a normal distribution function.

STEP 6

The area under the curve between Z23Z_{23}Z23​ and Z25Z_{25}Z25​ is the difference between the area to the left of Z25Z_{25}Z25​ and the area to the left of Z23Z_{23}Z23​.(23<X<25)=(Z25)−(Z23)(23 < X <25) =(Z_{25}) -(Z_{23})(23<X<25)=(Z25​)−(Z23​)

STEP 7

STEP 8

Plug in the values for the total number of days and the probability to calculate the number of days.Number of days=210times[(Z25)−(Z23)]Number\, of\, days =210 \\times [(Z_{25}) -(Z_{23})]Numberofdays=210times[(Z25​)−(Z23​)]

STEP 9

The result should be rounded to the nearest whole number, as it is not possible to have a fraction of a day.

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Calculate the Z-score for23 minutes.
$Z_{23} = \frac{23 -28}{3}$

Calculate the Z-score for25 minutes.
$Z_{25} = \frac{25 -28}{3}$

The area under the curve between $Z_{23}$ and $Z_{25}$ is the difference between the area to the left of $Z_{25}$ and the area to the left of $Z_{23}$ . $(23 < X <25) =(Z_{25}) -(Z_{23})$

Plug in the values for the total number of days and the probability to calculate the number of days.
$Number\, of\, days =210 \\times [(Z_{25}) -(Z_{23})]$