A company installs 5000 light bulbs, each with an average life of 500 hours, standard deviation of 100 hours, and distribution approximated by a normal curve. How many bulbs should have a life expectancy between 400 and 600 hours?
If you graph this, you will see a bell curve where the center of the bell curve, the mean = 500 hrs. Light bulbs with life expectancy within +/-1 SD will represent 68.27% of the total of all bulbs. The remainder, 31.73% of all light bulbs will have either longer or shorter life expectacy. Half of the remainder, or 15.865% will be longer lived (greater than 600 hrs) , half shorter (less than 400 hrs).
So the total number of bulbs with life expectancy greater than 400 hrs is all bulbs within the +/-1 SD range plus all bulbs over the 600 hr range.
Number = 0.6827 * 5000 + 0.15865 * 5000 = 4206 bulbs
That is not one of the answers =(
Why didn't you post the choices? Could this be a significant figure problem; i.e., round that 4206.
In the old days, we solved these "normal distribution" questions by using normal distribution tables, found in the back of text books.
Most calculators can now do these type of questions, after setting them in statistics mode.
However, there are several on-line webpages that simulate those charts.
One of the best I found is here:
http://davidmlane.com/normal.html
Just enter the mean and SD, then click on <between> and enter 400 and 600
I get 0.6827
so .6827(5000) = 3413.5 or 3414 to the nearest bulb
I see that Gabby did have 68.27% as the range, but don't understand why she did not use that probability .
choices:
1. 5000
2. 4750
3. 2500
4. 3400
3414 vs 3400 ... mhhh, what do you think?
yes, I got that answer, I figured out each using an area table
500-400=100
100/100=1
500-600=-100
-100/100=-1
on area table 1.00 = .341
.341+.341= .682 rounded =.68
.68*5000=3400
There you go!
good job
To determine the number of bulbs with a life expectancy between 400 and 600 hours, we need to calculate the area under the normal distribution curve within this range. This can be done using z-scores.
1. Calculate the z-score for the lower value of 400 hours:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
z = (400 - 500) / 100
z = -1
2. Calculate the z-score for the upper value of 600 hours using the same formula:
z = (600 - 500) / 100
z = 1
3. Look up the corresponding area under the normal distribution curve for these z-scores. You can use a z-score table or a statistical calculator to find these values. The area between -1 and 1 is approximately 0.6827.
4. To find the number of bulbs within this range, calculate the percentage and multiply it by the total number of bulbs:
Percentage = Area * 100
Number of bulbs = Percentage * Total number of bulbs
Number of bulbs = 0.6827 * 5000
Number of bulbs = 3413.5
Since you can't have half a bulb, you should round the result to the nearest whole number. Therefore, there should be approximately 3414 bulbs with a life expectancy between 400 and 600 hours.