Open ended question

Question

Assume that the heights of women are normally distributed with a mean of 64.1 inches and a standard deviation of 2.3 inches . If 64 women are randomly selected, find the probability that they have a mean height greater than 63.5’inches

Answer 1

Sabrina, it is important to understand the relationship between the standard deviation and the "Standard Error". Simplifying, the standard deviation is the amount of variability of one data point within a sample or a population; in the case above, it helps us determine the probability of how far under, or above the mean (average) we are likely to find a data point. If I pick two different data points, they are likely to be different. How different is defined by the standard deviation.

The Standard Error is nothing else than the standard deviation for a sample set with more that one data point in my set. It is the amount of variability of the sample mean (average of all the data in my sample) to be above or below the population mean. You can see that the Standard Error depends on the sample size; the larger the sample, the smaller the Standard Error; as I increase the number of measurements, the average of my sample is likely to get very close to the true population mean; and if I sampled my entire population, then my Standard Error (for that super large sample size) would be 0 (I know for sure what my sample mean is since I measured it, and if I were to sample my entire population again and again, the sample mean will always be the same. Conversely, if I sampled only one data point, my Standard Error for that single data sample would be equal to my population standard deviation.

Knowing the value of the Standard Error allows us to make inferences about the probability of a sample mean (of a given size) to be within a certain number of Standard Errors around the population mean.

The relationship between the Standard Error (SE) and a population standard deviation σ is : SE = σ / √n, with n being the sample size. In our case, the Standard Error would be: SE = 2.3/SQRT(64) = 0.288. Again, think of the Standard Error being the standard deviation for the sample mean. Notice that my Standard Error is way smaller than my population standard deviation; this is because my sample size is already quite significant (64 women). The mean of a sample of that size should not vary too much away from the population mean. We can now calculate the probability of being above or below that value.

Excel has allowed us to cheat a little and not have to do too much heavy math or looking up any tables to calculate that probability; we can use the Excel function p=NORM.DIST(x,mean,std dev, TRUE) to calculate the probability of a value to be less than x given a mean μ and a standard deviation σ.

In our case, we simply substitute the values for: p=NORM.DIST(63.5,64.1,0.288,TRUE) = 0.018.

This means there is a 1.8% chance the sample mean would be lower than or equal to 63.5 inches.

Therefore, there is a 100-1.8 = 98.2% chance the sample mean is greater than 63.5 inches, given the population mean and standard deviation we started with.

You could have also standardized the values by reducing the values to a standard normal distribution. In this case, we express the distance between two mean measurements in number of Standard Errors; for a standard normal distribution, this is expressed in a Z score. The Z score for a sample mean is simply the number of Standard Errors away from the population mean. Say X _bar is my sample mean (63.5 inches in our case). Then, Z = (X_bar-μ)/SE; this Z-value or Z-score will tell you in Standard Error units how far your sample mean is from the population mean. Z = (63.5-64.1)/0.288 = -2.087. In Standard Error units, that's a long way to the left of the mean. Looking at the standard normal curve, you will find yourself sitting on the far left tail.

The probability for a value X to be lower than Z given μ =1, and σ=0 is: p=NORM.S.DIST(Z,TRUE). Notice we don't need the real values anymore, this is because we reduced our distribution to the standard normal one. p= NORM.S.DIST(-2.087,1) = 0.018.

We find exactly the same result as above! the probability of being more than 2.087 standard errors below the mean (for a standard normal distribution) is 0.018.

So again, the probability of being greater than -2.087 is 98.2%.

Hope this helps!