The amount of time a certain brand of light bulb lasts is normally distribued with a mean of 1300 hours and a standard deviation of 60 hours. Out of 710 freshly installed light bulbs in a new large building, how many would be expected to last between 1240 hours and 1430 hours, to the nearest whole number?
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davidmlane.com/hyperstat/z_table.html
To find out how many light bulbs would be expected to last between 1240 and 1430 hours, we need to calculate the area under the normal distribution curve within this range.
First, let's calculate the z-scores for each of the values:
For 1240 hours:
z1 = (1240 - 1300) / 60 = -1
For 1430 hours:
z2 = (1430 - 1300) / 60 = 2.1667 (rounded to 4 decimal places)
Next, we will use a standard normal distribution table or calculator to find the area under the curve between these z-scores.
Using the table or calculator, find the area corresponding to z1 and subtract it from the area corresponding to z2:
P(z1 < Z < z2) = P(-1 < Z < 2.1667)
This gives us the proportion of the light bulbs expected to fall within the range of 1240 to 1430 hours.
Finally, multiply this proportion by the total number of light bulbs to get the expected number of light bulbs that fall within this range.
Let's calculate the expected number of light bulbs:
P(z1 < Z < z2) = P(-1 < Z < 2.1667) = 0.8659
Expected number of light bulbs = 0.8659 * 710 = 614.09
Rounded to the nearest whole number, we can expect approximately 614 light bulbs to last between 1240 and 1430 hours.
To find out how many light bulbs would be expected to last between 1240 hours and 1430 hours, we need to use the properties of the normal distribution.
First, let's calculate the z-scores for the lower and upper limits of the range.
Z-score = (X - μ) / σ
Where:
X = Value of interest (lower or upper limit)
μ = Mean of the distribution (1300 hours)
σ = Standard deviation of the distribution (60 hours)
For the lower limit (X = 1240 hours):
Z-score = (1240 - 1300) / 60
Z-score = -60 / 60
Z-score = -1
For the upper limit (X = 1430 hours):
Z-score = (1430 - 1300) / 60
Z-score = 130 / 60
Z-score ≈ 2.17
Next, we need to find the areas under the normal distribution curve corresponding to these z-scores.
You can either use statistical tables or a calculator that has a built-in normal distribution function. For simplicity, let's use an online calculator.
Using the online calculator, the area under the normal distribution curve to the left of -1 is approximately 0.1587. This implies that about 15.87% of the bulbs will last less than 1240 hours.
The area under the normal distribution curve to the left of 2.17 is approximately 0.9857. This implies that about 98.57% of the bulbs will last less than 1430 hours.
To find the proportion of bulbs that will last between 1240 and 1430 hours, we subtract the area to the left of the lower limit (15.87%) from the area to the left of the upper limit (98.57%):
Proportion = 0.9857 - 0.1587 ≈ 0.8270
Finally, we multiply this proportion by the total number of light bulbs (710) to get the expected number of light bulbs that will last between 1240 and 1430 hours:
Expected number = Proportion * Total number
Expected number ≈ 0.8270 * 710
Rounding this result to the nearest whole number, we find that approximately 588 light bulbs would be expected to last between 1240 hours and 1430 hours.