Answer:
a)3.438% of the light bulbs will last more than 6262 hours.
b)11.31% of the light bulbs will last 5252 hours or less.
c) 23.655% of the light bulbs are going to last between 5858 and 6262 hours.
d) 0.12% of the light bulbs will last 4646 hours or less.
Step-by-step explanation:
Normally distributed problems can be solved by the z-score formula:
On a normaly distributed set with mean and standard deviation
, the z-score of a value X is given by:
After we find the value of Z, we look into the z-score table and find the equivalent p-value of this score. This is the probability that a score will be LOWER than the value of X.
In this problem, we have that:
The lifetimes of light bulbs are approximately normally distributed, with a mean of 5656 hours and a standard deviation of 333.3 hours.
So
(a) What proportion of light bulbs will last more than 6262 hours?
The pvalue of the z-score of is the proportion of light bulbs that will last less than 6262. Subtracting 100% by this value, we find the proportion of light bulbs that will last more than 6262 hours.
has a pvalue of .96562. This means that 96.562% of the light bulbs are going to last less than 6262 hours. So
of the light bulbs will last more than 6262 hours.
(b) What proportion of light bulbs will last 5252 hours or less?
This is the pvalue of the zscore of
has a pvalue of .1131. This means that 11.31% of the light bulbs will last 5252 hours or less.
(c) What proportion of light bulbs will last between 5858 and 6262 hours?
This is the pvalue of the zscore of subtracted by the pvalue of the zscore
For , we have that
with a pvalue of .96562.
For X = 5858
has a pvalue of .72907.
So, the proportion of light bulbs that will last between 5858 and 6262 hours is
23.655% of the light bulbs are going to last between 5858 and 6262 hours.
(d) What is the probability that a randomly selected light bulb lasts less than 4646 hours?
This is the pvalue of the zscore of
has a pvalue of .0012. This means that 0.12% of the light bulbs will last 4646 hours or less.