# Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inches. If one male is randomly selected, what is the probability that his head breadth is less than 6.2 inches?

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The Safeguard Helmet Company plans an initial production run of 1200 helmets. What is the probability that 100 randomly selected men have a mean head breadth less than 6.2 inches?

The Safeguard Helmet Company plans an initial production run of 1200 helmets. What is the probability that 100 randomly selected men have a mean head breadth less than 6.2 inches?

##### 1 Answer

.5793 or 57.93%

.9772 or 97.72%

#### Explanation:

We want to find the probability that, in a normally distributed series, we will encounter a value that is less than the value that is .2 greater than the mean.

The z-table is great for this kind of problem.

The z-score for a head breadth of 6.2 would be .2 because the mean is 6, and the standard deviation is 1.

We can now look in our positive z-table (the one I've linked here is from chegg.com) to find the area of a normal curve to the left of

**Edit:** I see that this is a two-part question.

For the second part, you want to use this formula:

with

We can now use the central limit theorem.

Going back to our z table:

or a 97.72% chance.