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In a sample of adult women from the United States, the average height was 164.4 cm and the standard deviation was 6.2 cm. Women who are more than 2 standard deviations above the mean are considered very tall, and women who are more than two standard deviations below the mean are considered very short. Height in women is normally distributed.
a. What are the heights of very tall and very short women?
b. In a population of 10,000 women, how many are expected to be very tall and very short?

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Answer:

176.8 cm and 152 cm

420

Step-by-step explanation:

[tex]\mu[/tex] = Mean = 164.4 cm

[tex]\sigma[/tex] = Standard deviation = 6.2

[tex]\mu-2\sigma\\ =164.4-2\times 6.2=14.4\\ =152\ cm[/tex]

[tex]\mu+2\sigma\\ =164.4+2\times 6.2=14.4\\ =176.8\ cm[/tex]

The heights of very tall women are 176.8 cm and very short are 152 cm

The percentage of tall women is 2.1 %

[tex]0.021\times 10000=210[/tex]

The percentage of short women is 2.1 %

[tex]0.021\times 10000=210[/tex]

So, 420 are expected to be very tall and very short