Question

section 3.5 spotlight on statistics
objective 1 - understand standard error of the mean
guided practice activity #11 - z - scores and standard error
the histograms we saw above are discrete versions of idealized continuous distributions, such as
the following normal distribution:

1. in the normal distribution what percentage of data values fall below (to the left of) the mean,

μ?
below + 1σ(=+1 standard deviation)?
below + 2σ?

Explanation:

Step1: Analyze data below the mean

The normal distribution is symmetric about the mean. The total area under the curve is \(100\%\). Since it is symmetric, the area to the left of the mean (\(\mu\)) is \(50\%\).

Step2: Analyze data below \(+ 1\sigma\)

The area to the left of the mean (\(50\%\)) plus the area between the mean and \(+1\sigma\) (\(34.13\%\)) gives \(50\%+34.13\% = 84.13\%\)

Step3: Analyze data below \(+2\sigma\)

The area to the left of the mean (\(50\%\)) plus the area between the mean and \(+1\sigma\) (\(34.13\%\)) plus the area between \(+1\sigma\) and \(+2\sigma\) (\(13.59\%\)) gives \(50\%+34.13\%+13.59\%=97.72\%\)

Answer:

• Below the mean (\(\mu\)): \(50\%\)
• Below \(+1\sigma\): \(84.13\%\)
• Below \(+2\sigma\): \(97.72\%\)