Question

scientists measured the lengths of a species of bird in a region. the graph below shows that the measured lengths have a normal distribution. about what percent of the bird population is between 14 and 34 cm long?
a) 78% b) 98%
c) 68% d) 88%

1. a candy store owner wants to know the overall thoughts of customers in her store. she asks the first 20 custormoers who come into her store on a monday morning and ask them to rate there store experience on a scale of 1 to 10. her customers average response was a 8.8 and she concludes that all her customers have positive thoughts about her store. her conculsion is:

a) invalid, since the customers were chosen by convenience b) unfair because she owns the store
c) spot on! d) valid, since customers were chosen randomly

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\mu\pm\sigma\)) of the mean.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\mu\pm2\sigma\)) of the mean.
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\mu\pm3\sigma\)) of the mean.

From the graph, the mean \(\mu = 22\). The distance from the mean to \(14\) is \(22 - 14=8\), and the distance from the mean to \(34\) is \(34 - 22 = 12\). Wait, no, actually, if we assume that the intervals between the marks on the \(x -\)axis are equal. Let's assume the standard deviation \(\sigma=4\) (since \(22-14 = 8=2\sigma\) and \(34 - 22=12 = 3\sigma\) is wrong. Wait, no, if we consider that \(14=22 - 2\times4\) and \(34=22+3\times4\) is incorrect. Wait, actually, if we use the fact that in a normal distribution, the total area under the curve is \(1\) (or \(100\%\)). The value \(14\) is \(22- 8\) and \(34\) is \(22 + 12\) is wrong. Wait, no, looking at the normal - distribution curve:
The area between \(\mu - 2\sigma\) and \(\mu+3\sigma\). The area from \(\mu - 2\sigma\) to \(\mu\) is \(47.5\%\) (from the empirical rule: \(95\%\) within \(\mu\pm2\sigma\), so \(47.5\%\) on each side of the mean for \(\mu\pm2\sigma\)) and the area from \(\mu\) to \(\mu + 3\sigma\) is \(49.85\%\) (since \(99.7\%\) within \(\mu\pm3\sigma\), so \(49.85\%\) on each side of the mean for \(\mu\pm3\sigma\)). The total area between \(14\) (if \(\mu = 22\) and \(\sigma = 4\), \(14=22-2\times4\)) and \(34\) (\(34=22 + 3\times4\)) is \(47.5\%+49.85\%=97.35\%\approx98\%\)

Answer:

b) \(98\%\)