Answer:
- <u><em>The standard deviation of the distribution is 1.5ºF.</em></u>
Explanation:
The known 68-95-99.7 rule, the empirical rule, states that, in a normal distribution 68% of the data are within one standard deviation from the mean, 95% of the data are within two standard deviations from the mean, and 99.7% are within three standard deviations:
- 68% ⇒ mean ± 1 standard deviation
- 95% ⇒ mean ± 2 standard deviations
- 99.7% ⇒ mean ± 3 standard deviation.
Hence, for the<em> approximately normal distribution of the daily high temperatures</em>,<em> with mean 86º</em>, and a range of <em>83ºF to 89ºF</em> for <em>95% of the data,</em> you can write:
- 86ºF ± 2 SD ⇒ 86ºF - 2 SD = 83ºF, and
- 86ºF ± 2 SD ⇒ 86ºF + 2SD = 89 ºF
Both equations will lead to the same result:
- 86ºF - 2 SD = 83ºF ⇒ 2 SD = 86ºF - 83ºF = 3ºF
SD = 3ºF / 2 = 1.5ºF
Also:
- 86ºF + 2SD = 89ºF ⇒ 2 SD = 89ºF - 86ºF = 3ºF
SD = 3ºF / 2 = 1.5ºF
Therefore, the standard deviation of the distribution is 1.5ºF.
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