Probability for Normal Distribution

Question

Suppose that motorists in the southeastern United States use a mean of 8.20 gallons of gasoline per week with a standard deviation of 0.47 gallons. Assume that gasoline consumption levels are approximately normally distributed.

a. What percentage of southeastern drivers user no more than 7.0 gallons of gasoline per week?

b. What percentage of southeastern drivers use an amount of gasoline that differs from the mean by more than 0.5 gallons per week?

Answer 1

With your mean and standard deviation given, and since we are told to assume a normal distribution, we have enough information to find a z-score. A z-score means "how many standard deviations away from the mean is this?"

For the first part, we want to measure how many standard deviations 7.0 is away from the mean of 8.2. Since 1 standard deviation is 0.47, you might be already able to see that 7.0 is a bit more than 2 standard deviations below the mean. To get the exact value, take their difference and divide by the size of the standard deviation:

z = (7.0 - 8.2)/0.47 = -2.553

The negative here indicates that we are below the mean. It should be mentioned that any z value under -2 or over 2 will be quite rare.

We are asked what percentage of people are at or below this value of z = -2.55. This will be a single tail of the distribution, and since z = -2.55 is so far negative it will be a very small percentage. If your z-table only has positive values, you can use them as well since the normal distribution is symmetric. Just make sure you get the "big vs. small" right. The z-table will tell you what portion of the population is on one side or the other of that z-value (usually the left side). Here we are looking for a very small percentage. But if we look up positive z = 2.55, we see 0.9946 in the table. This means that 99.46% of people will have z-value smaller than z=2.55. Flip that around to the negative side: by the symmetry of the distribution it means 99.46% of people will have z-value larger than z=-2.55. We were looking for smaller, and so we just subtract 99.46% from 100% to get 0.44%. This is the small value we were expecting. So 0.44% is the percentage of people using no more than 7.0 gallons per week.

When you are still getting used to z-tables, it can be confusing as to when you subtract from 100% and when you don't. If you keep in mind "should this be big or small?", it hopefully will be less confusing.

For the second part, we are looking for people far from the mean on either side. So this will be two tails. Our z-value for "0.5 gallons away from the mean" is just 0.5/0.47 or z = 1.06. Looking that up in the z-table, we get 0.8554, or 85.54%. But that is the percentage to the left of that value, when we want both tails. Let's focus on one tail at a time. Tails are small, so subtract from 100% again to get the (single) tail:

100% - 85.54% = 14.46%

The other tail will also be 14.46% since the normal distribution is symmetric. So just double that percentage and you are done!