7.7 standard deviations away from the mean?

Question

I'm confused. I have a problem where I have to find the probability that x is below the z value 7.7. My z table only goes to z values of 3.4. How do I calculate this? These are the hints my teacher gave me...

A z-value of 7.7 means that we have a value that is 7.7 standard deviations away from the mean....you're not wrong here. Think about these questions: - Is this likely to happen? - What is the probability of having a value at less than 7.7 standard deviations away from the mean?

Thanks!

Answer 1

The probability that a random variable $Z$ with standard normal distribution is less than $7.7$ is, for all practical purposes, equal to $1$. We have $$\Pr(Z \gt 7.7)\approx 6.8\times 10^{-15}.$$

The probability that we are $7.7$ or more standard deviations away from the mean (either direction allowed) is twice that. But twice utterly negligible is still utterly negligible.

A look at the graph of the graph of the characteristic "bell-shaped" density function of the standard normal shows that almost all the area is concentrated between $-3.5$ and $3.5$.

Remark: Suppose that you buy a single ticket in one of the mega-million lotteries this year, and again a single ticket next year. The probability that you will be the grand prize winner both times is greater than $\Pr(Z\gt 7.7)$.

Answer 2

A slight overestimate of, and a reasonable approximation to, $P(Z > 7.7)$ is obtained by

1. Calculating the value of the standard normal density function (yes, I did mean to write density function) at $7.7$. $$\frac{1}{\sqrt{2\pi}}e^{-7.7^2/2} \approx 5.32\times 10^{-14}$$

2. Divide the result by $7.7$ $$\frac{5.32\times 10^{-14}}{7.7} \approx 6.909\times 10^{-15}$$

which can be compared to the more exact result of $6.8\times 10^{-15}$ given by André.

Answer 3

The probability of a normally distributed random variable being within 7.7 standard deviations is practically 100%.

Remember these rules: 68.2% of the probability density is within one standard deviation; 95.5% within two deviations, and 99.7 within three deviations.

The reason that tables don't go to 7.7 is because deviations beyond around three are of little practical use. If you see a random variable being out by 7.7 deviations, this is so unlikely that you should suspect something is wrong with the experiment.

To calculate the exact answer, you have to simply figure out the area under the bell curve between 0 and 7.7 and then multiply by two. To do that, you need the cumulative density function: i.e., the integral of the probability density function. Unfortunately, that function does not exist in closed form.

Since you're expected to compute this for homework, your teacher must have given you some tools by which he or she expects you to calculate cumulative densities that are not covered by your table.

Here is a paper about approximating the cumulative density function.

Scientific calculators which provide support for statistical computing often have a function for this. You enter an argument like 7.7, and the function computes the cumulative density from $-\infty$ to the argument: i.e. the area under the curve to the left of the argument. If you have such a function, then simply get the value for 7.7, and then subtract from that the value for -7.7.

Your calculator most likely performs this calculation using numerical integration over the probability density function, rather than an approximation formula.