Shown below are the data giving the length of life of lights in an underground tunnel.

9.0 months 7.0 months
8.2 months 8.3 months
7.8 months 8.1 months
7.5 months 7.6 months
9.1 months 7.4 months

I calculated the standard deviation to be about 0.679.
Estimate the probability that a random chosen light will last as long as 7 months after installation. Assume the data to be described by a normal distribution.

Can anyone help me with the process I would use to calculate the answer? Thanks!

Use z-scores. Find the mean. You have already calculated standard deviation.

Formula for z-scores is the following:
z = (x - mean)/sd

Use 7 for x. Calculate z-score. Determine the probability using a z-distribution table.

Thanks for the insight!

So I've calculated the z-score to be about -1.47, or about -1.5. Looking at a z-distribution table, I see on the left a column for z-scores and I see -1.5 there, but there are many values corresponding on the right?

Do you know how I know which one to use?

Never mind- I've figured it out, THANK YOU MathGuru!

To estimate the probability that a randomly chosen light will last as long as 7 months after installation, you can use the concept of a standard normal distribution.

First, you need to calculate the z-score for the value of 7 months. The z-score is a measure of how many standard deviations a particular value is away from the mean. It can be calculated using the formula:

z = (x - μ) / σ

where x is the value of interest (7 months in this case), μ is the mean of the distribution, and σ is the standard deviation.

In this case, you mentioned that the standard deviation is approximately 0.679. However, the mean is not provided. To estimate the mean, you can calculate the average of all the given data points:

mean = (9.0 + 7.0 + 8.2 + 8.3 + 7.8 + 8.1 + 7.5 + 7.6 + 9.1 + 7.4) / 10 = 8.0

Now you can calculate the z-score:

z = (7 - 8.0) / 0.679 = -1.475

The next step is to use a z-table (also known as a standard normal distribution table) to find the corresponding probability. The z-table provides the cumulative probability up to a given z-score.

Looking up a z-score of -1.475 in the table, you can find the area under the curve to the left of that z-score. Subtracting this area from 0.5 (the total area under the curve) will give you the probability that a randomly chosen light will last as long as 7 months or less after installation.

Keep in mind that the table might provide a standard normal distribution table for positive z-scores only. In that case, you can use the symmetry of the distribution to find the equivalent positive z-score for -1.475 and then look up the corresponding probability.

I hope this explanation helps you in calculating the probability!