When Dominic commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 26 minutes and a standard deviation of 3 minutes. Out of the 269 days that Dominic commutes to work per year, how many times would his commute be between 28 and 31 minutes, to the nearest whole number?
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To find out how many times Dominic's commute would be between 28 and 31 minutes, we need to calculate the area under the normal distribution curve within that range.
First, we need to standardize the values of 28 and 31 using the z-score formula:
z = (x - μ) / σ
where:
x = value we want to standardize (in this case, 28 and 31)
μ = mean of the distribution (26 minutes)
σ = standard deviation of the distribution (3 minutes)
So, for 28 minutes:
z1 = (28 - 26) / 3 = 2/3
And for 31 minutes:
z2 = (31 - 26) / 3 = 5/3
We can use a standard normal table or a calculator to find the area under the curve between these two z-scores.
Using a standard normal table, the area between 2/3 and 5/3 is approximately 0.18593.
Finally, we need to multiply this area by the total number of days Dominic commutes to work per year to get the approximate number of times his commute would be between 28 and 31 minutes:
Number of times = 0.18593 * 269 ≈ 49.97
Rounding to the nearest whole number, Dominic's commute would be between 28 and 31 minutes approximately 50 times per year.
To find the number of times Dominic's commute would be between 28 and 31 minutes, you need to calculate the probability that his commute time falls within that range.
First, let's calculate the z-scores for the lower and upper limits of the range. The z-score is a measure of how many standard deviations an observation is from the mean and helps in converting values to a standard normal distribution.
For the lower limit (28 minutes):
z = (x - μ) / σ
z = (28 - 26) / 3
z = 2 / 3 ≈ 0.67
For the upper limit (31 minutes):
z = (31 - 26) / 3
z = 5 / 3 ≈ 1.67
Next, we need to find the proportion (or probability) of the commute time falling within the range of 28 to 31 minutes. This can be done by finding the area under the normal curve between these z-scores.
Using a z-table or a statistical calculator, you can find the probabilities corresponding to the lower and upper z-scores. Subtracting the lower probability from the upper probability will give you the proportion of the area under the curve within that range.
Assuming a normal distribution table or calculator, the probability corresponding to a z-value of 0.67 is approximately 0.7486, and the probability corresponding to a z-value of 1.67 is approximately 0.9525.
So, the proportion of commute times between 28 and 31 minutes is:
0.9525 - 0.7486 = 0.2039
Now, to find the number of times his commute would be within that range out of 269 days, you can multiply the proportion by the total number of days:
0.2039 * 269 ≈ 54.94
Rounding to the nearest whole number, Dominic's commute would be between 28 and 31 minutes approximately 55 times per year.