Suppose that the lifetime of a farm tractor follows a normal distribution with a mean of eight years and a standard deviation of 1.7 years. If the manufacturer is willing to replace only 5% of the tractors that fail within the warranty period, how long should the warranty period be?
http://davidmlane.com/hyperstat/z_table.html
Answer and solution pleaseee
Hey, just try it
Value from area circle
area = .95 (which is 1 - .05)
mean = 8
sd = 1.7
then black area (.95 still working) is below 10.797 years
Well, if the manufacturer is only willing to replace 5% of the tractors that fail within the warranty period, they might want to consider renaming themselves to the "We-Don't-Cover-You Manufacturing Company." But hey, let's crunch some numbers for fun!
In order to figure out how long the warranty period should be, we need to determine the cutoff point for when a tractor is considered a failure. Since the manufacturer is only replacing 5% of the tractors, we need to find the 5th percentile of the normal distribution.
To find the 5th percentile, we can use a standard normal table or a statistical calculator. Plugging in the mean (8 years) and the standard deviation (1.7 years), we can find that the 5th percentile is approximately 6.4 years.
So, if the manufacturer wants to replace only 5% of the tractors that fail within the warranty period, the warranty period should be set to 6.4 years. Just enough time to start placing bets on how long your tractor will last!
To determine how long the warranty period should be, we need to find the value of the lifetime of the tractor that corresponds to the 5th percentile of the normal distribution.
Step 1: Convert the percentile to a z-score.
Since we are looking for the 5th percentile, the area under the normal curve to the left of the z-score is 0.05. We can use a standard normal distribution table or a calculator to find the z-score corresponding to this area. In this case, the z-score is approximately -1.645.
Step 2: Use the z-score formula to find the corresponding value in the original units.
The z-score formula is:
z = (x - μ) / σ
where z is the z-score, x is the value in the original units, μ is the mean, and σ is the standard deviation.
Rearranging the formula to solve for the value x:
x = z * σ + μ
Substituting the values, we have:
x = -1.645 * 1.7 + 8
Calculating the value, we find:
x ≈ 4.745
Step 3: Round up to the nearest whole number.
Since the tractor lifetime cannot be a fraction of a year, we round up the value to the nearest whole number.
Therefore, the warranty period should be approximately 5 years.