How do I solve normal distribution word problems? Ex
A manufacturer offers a warranty on it's coffee makers. The coffee makers have mean lifespan of 4.5 years, with standard deviation of 0.7 yeaes. For how long should the coffee makers be covered by the warranty, if the manufacturer wants to repair no more than 2.5% of the coffee makers sold?
To solve this problem, you can use the concept of the standard normal distribution.
Step 1: Convert the problem to the standard normal distribution by using the z-score formula:
z = (x - μ) / σ
where x is the value you are interested in (in this case, the lifespan of the coffee makers), μ is the mean, and σ is the standard deviation.
Step 2: Find the z-score corresponding to the desired percentage. In this case, we are interested in the lower tail probability of 2.5%. So, we want to find the z-score such that P(Z ≤ z) = 0.025. You can use a standard normal distribution table or a calculator to find the corresponding value.
Step 3: Once you have found the z-score, you can use the formula z = (x - μ) / σ to solve for x, the lifespan of the coffee makers. Rearrange the formula to solve for x:
x = z * σ + μ
where x is the lifespan of the coffee makers, z is the z-score corresponding to the desired percentage, σ is the standard deviation, and μ is the mean.
Step 4: Plug in the values for the mean and standard deviation from the problem statement and the z-score you found in Step 2 into the formula you derived in Step 3. Solve for x.
So, in this problem, to determine the duration of the warranty, we need to find the value of x (lifespan of the coffee makers) such that the probability of a coffee maker failing before x is no more than 2.5%.
To solve normal distribution word problems like the one you mentioned, you can follow the steps below:
Step 1: Understand the problem
Read the problem carefully and identify the relevant information. In this case, we are given the mean lifespan of the coffee makers (4.5 years), the standard deviation (0.7 years), and the manufacturer's requirement regarding the percentage of coffee makers to be repaired (2.5%).
Step 2: Determine the z-score
The z-score measures the number of standard deviations an observation is from the mean. It helps us reference a specific value in the normal distribution.
To find the z-score, use the following formula:
z = (x - μ) / σ
where:
x = the value you want to find the probability for (in this case, the lifespan below which the manufacturer wants to repair no more than 2.5%)
μ = the mean lifespan (4.5 years)
σ = the standard deviation (0.7 years)
Since we want to find the lifespan below which the manufacturer wants to repair no more than 2.5%, we need to find the z-score for the upper 2.5% tail.
Step 3: Find the z-value
Using a standard normal distribution table or a statistical calculator, find the z-value corresponding to the upper 2.5% tail. In this case, you are looking for the z-value which gives a cumulative probability of 0.025 in the upper tail.
Once you find the z-value, note that it represents the number of standard deviations away from the mean.
Step 4: Calculate the value of interest
Using the z-value, the mean, and the standard deviation, you can calculate the value of interest. Rearranging the z-score formula:
x = z * σ + μ
Substituting the z-value, mean, and standard deviation, you can calculate the lifespan below which the manufacturer wants to repair no more than 2.5%.
So, to find the answer to the specific problem you provided, follow these steps.
you want x such that P(Z<x) = 0.025
That means that x must be more than 1.96 std below the mean. In this case, that's 3.128 years.