A certain type of lightbulb is advertised to have an average lifetime of 1,000 hours. Assume the lifetimes of these lightbulbs are approximately normally distributed with a standard deviation of 250 hours.

A) Find the percentage of lightbulbs that will fail during their first 250 hours of use
B) Find the percentage of lightbulbs that will fail within 100 hours of their average lifetime (either 50 hours more or less than average).
C) At what number of hours should the warranty lifetime be set so that fewer than 2% of the bulbs must be replaced under warranty?

Not sure what formulas to use in excel to calculate these answers. Norm.dist or Norm.Inv

Z = (score-mean)/SD

A,B. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores. Multiply by 100.

C. Use same table to find Z for .02. Insert in above equation to find score.

To answer these questions using Excel, you can use the NORM.DIST and NORM.INV functions. Here's the step-by-step breakdown for each question:

A) Find the percentage of lightbulbs that will fail during their first 250 hours of use:

1. Open Excel and enter the following values in separate cells:
- Average lifetime (mean): 1000
- Standard deviation: 250
- Time threshold: 250

2. In an empty cell, enter the formula: =NORM.DIST(Time threshold, Average lifetime, Standard deviation, TRUE)
- This function calculates the cumulative probability of a value being less than or equal to the Time threshold, given the mean and standard deviation.
- The TRUE argument signifies that the function returns the cumulative probability.

3. Press Enter to get the result, which will be the percentage of lightbulbs that will fail during their first 250 hours of use.

B) Find the percentage of lightbulbs that will fail within 100 hours of their average lifetime:

1. Open Excel and enter the following values in separate cells:
- Average lifetime (mean): 1000
- Standard deviation: 250
- Time threshold: 100

2. In an empty cell, enter the formula: =NORM.DIST(Time threshold, Average lifetime, Standard deviation, TRUE)
- This function calculates the cumulative probability of a value being less than or equal to the Time threshold, given the mean and standard deviation.
- The TRUE argument signifies that the function returns the cumulative probability.

3. Subtract the result from step 2 from 0.5 (representing 50%) since you want the combined probability for both less and more than the average lifetime.
- In an empty cell, enter the formula: =0.5 - result_from_step_2
- Replace "result_from_step_2" with the cell reference from step 2.

4. Multiply the result from step 3 by 100 to get the percentage of lightbulbs that will fail within 100 hours of their average lifetime.

C) At what number of hours should the warranty lifetime be set so that fewer than 2% of the bulbs must be replaced under warranty:

1. Open Excel and enter the following values in separate cells:
- Average lifetime (mean): 1000
- Standard deviation: 250
- Desired warranty failure rate: 2% (0.02)

2. In an empty cell, enter the formula: =NORM.INV(1 - Desired warranty failure rate, Average lifetime, Standard deviation)
- This function calculates the value that corresponds to the given cumulative probability, given the mean and standard deviation.
- The 1 - Desired warranty failure rate is used to find the corresponding value from the upper tail of the distribution.

3. Press Enter to get the result, which will be the number of hours at which the warranty lifetime should be set so that fewer than 2% of the bulbs must be replaced under warranty.

By following these steps, you can use Excel to calculate the answers to the provided questions.

To answer these questions using Excel, you can indeed use the functions Norm.Dist and Norm.Inv. Here's how you can use these functions to calculate the desired percentages and find the warranty lifetime:

A) To find the percentage of lightbulbs that will fail during their first 250 hours of use, you need to find the cumulative distribution function (CDF) of the normal distribution. Use the following formula in Excel:

`=NORM.DIST(250,1000,250,TRUE)`

The three arguments of the function are:
- 250: The value at which you want to evaluate the CDF.
- 1000: The mean of the distribution.
- 250: The standard deviation of the distribution.
- TRUE: Indicates that you want the cumulative probability from negative infinity up to the given value.

This will return the percentage of lightbulbs that fail during their first 250 hours of use.

B) To find the percentage of lightbulbs that will fail within 100 hours of their average lifetime, you need to find the difference between the average lifetime and 100 hours, and then find the CDF at that value. Use the following formula in Excel:

`=NORM.DIST(1000-100,1000,250,TRUE)`

This will return the percentage of lightbulbs that fail within 100 hours of their average lifetime.

C) To find the number of hours to set the warranty lifetime so that fewer than 2% of the bulbs must be replaced under warranty, you need to find the inverse of the cumulative distribution function (CDF). Use the following formula in Excel:

`=NORM.INV(0.02,1000,250)`

The two arguments of the function are:
- 0.02: The desired percentage (2%) as a decimal.
- 1000: The mean of the distribution.
- 250: The standard deviation of the distribution.

This will return the number of hours at which the warranty lifetime should be set.

Using these formulas, you can calculate the desired percentages and find the warranty lifetime in Excel.