if the life in years of a washing machine is normally distributed with a mean of 15 years and a standard deviation of 1.6 years what should be the guarantee period if the company wants less than 1% of the machines to fail while under warranty
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http://davidmlane.com/hyperstat/z_table.html
To calculate the guarantee period, we need to find the value that represents the cutoff point for the lowest 1% of the machines. This value is called the lower critical value.
1. First, we need to find the z-score corresponding to the desired probability of 1%. We can use the standard normal distribution table or a statistical calculator for this.
The z-score corresponding to a probability of 1% is approximately -2.33.
2. Now, we can use the formula for z-score to find the corresponding raw score or guarantee period:
z = (x - μ) / σ
Rearranging the formula to solve for x, we have:
x = μ + z * σ
where:
x = the guarantee period we want to find
μ = mean life of the washing machine = 15 years
σ = standard deviation of the life of the washing machine = 1.6 years
z = z-score for a 1% probability, which is -2.33 (obtained earlier)
3. Substituting the given values into the formula, we can calculate the guarantee period:
x = 15 + (-2.33) * 1.6
x ≈ 15 - 3.728
x ≈ 11.272
Therefore, the guarantee period for the washing machine should be approximately 11.272 years to ensure that less than 1% of the machines fail while under warranty.
To determine the guarantee period, we need to find the value at which less than 1% of the machines will fail while still under warranty. Here's how you can calculate it:
Step 1: Convert the problem into a standard normal distribution.
To do this, we'll use the formula:
Z = (X - μ) / σ
where Z is the standard score (z-score), X is the variable value, μ is the mean, and σ is the standard deviation.
Step 2: Find the z-score corresponding to the 1% failure rate.
Since we want less than 1% of the machines to fail, we need to find the z-score that corresponds to the cumulative probability (area under the curve) of 0.01. We can look up this value in the standard normal distribution table or use a calculator.
Step 3: Solve for X using the z-score.
Using the formula from Step 1, we can rearrange it to solve for X. Once we have the X-value, we can interpret it as the guarantee period.
Let's calculate the guarantee period:
Step 1: Convert the problem into a standard normal distribution.
Since the mean (μ) is 15 years and the standard deviation (σ) is 1.6 years, we have X = guarantee period, μ = 15 years, and σ = 1.6 years.
Step 2: Find the z-score corresponding to the 1% failure rate.
We need to find the z-score that has a cumulative probability of 0.01. Using a standard normal distribution table, we can find that the z-score is approximately -2.33.
Step 3: Solve for X using the z-score.
Applying the formula:
-2.33 = (X - 15) / 1.6
Now, let's solve for X:
-2.33 * 1.6 = X - 15
-3.728 = X - 15
X = 11.272
Therefore, the guarantee period should be rounded up to 12 years to ensure that less than 1% of the machines fail while under warranty.