Use these parameters (based on Data Set 1 "Body Data" in...

Use these parameters (based on Data Set 1 "Body Data" in Appendix $B$ ): Men's heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in. Women's heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in. Navy Pilots The U.S. Navy requires that fighter pilots have heights between 62 in. and 78 in. a. Find the percentage of women meeting the height requirement. Are many women not qualified because they are too short or too tall? b. If the Navy changes the height requirements so that all women are eligible except the shortest $3 \%$ and the tallest $3 \%,$ what are the new height requirements for women?

Use these parameters (based on Data Set 1 "Body Data" in Appendix $B$ ):
Men's heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.
Women's heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.
Navy Pilots The U.S. Navy requires that fighter pilots have heights between 62 in. and 78 in.
a. Find the percentage of women meeting the height requirement. Are many women not qualified because they are too short or too tall?
b. If the Navy changes the height requirements so that all women are eligible except the shortest $3 \%$ and the tallest $3 \%,$ what are the new height requirements for women?

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is defined by its mean and standard deviation, and a large number of natural and social phenomena can be modeled using this distribution.

Z-score and Standardization

A z-score measures how many standard deviations an element is from the mean. Standardization using z-scores allows any value from a normal distribution to be transformed into a common scale, enabling comparisons across different normal distributions and making it easier to compute probabilities and percentiles.

Percentile Calculation

Percentiles are used to understand the relative standing of a value within a dataset. In the context of a normal distribution, calculating percentiles involves determining the corresponding z-score and then using the z-score to find the cumulative probability up to that point. This is particularly useful for identifying thresholds such as the lowest or highest percentages in the data.

Tail Probabilities and Range Requirements

Tail probabilities refer to the portions of the distribution that lie beyond a specific cutoff point, typically at the extremes. In practical applications, such as setting qualification criteria, these tail probabilities can define exclusion zones, ensuring that only individuals within a certain middle range (or percentile interval) are eligible, while those in the tails (for instance, the shortest or tallest percentages) are excluded.

Transcript

00:02 Right, so according to the problem, we're looking at female's height here.

00:14 Okay, so the me is equal to 63 .7.

00:25 So, 63 .7 and we have standard deviation that is equal to 2 .9.

00:34 Now we're as if we have a raw score, the percentage of the female pilots having a high between 62 and 78.

00:54 So if we're looking at the standard normal curve, assuming that the data is normal, we're actually looking for the area under this curve.

01:06 That's 62 to 78.

01:10 So in order to do that, the most standardize our x values from x to z.

01:17 Okay, so we use the formula z equals x minus mu all over sigma.

01:26 So if the x is 62, we have z equals 62 minus 63 .7 over 2 .9.

01:38 Okay, and then if d x is equal to 7 to 8, we have 7 to 8 minus 63 .7 over 2 .9.

01:48 So we just have to calculate these values.

01:53 Okay, so allow me to calculate.

02:05 So calculating that we have a z score for z score that's right here, negative 0 .5816.

02:19 And then another z, okay.

02:31 So another z that's equal to 4 .9310.

02:40 Okay.

02:41 So in a standard normal curve, we're actually looking for the probability that the z is between negative 0 .5862 to 4 .9310.

02:56 Okay, so in order to do that, we have to take note that this probability is equal to the probability that z is less than 4 .0.

03:12 Sorry.

03:16 Z is less than 4 .930 minus the probability that z is less than negative 0 .5862.

03:29 Okay, so just have to look at the z table in order to get its values.

03:43 Okay, let me scroll this one out.

03:46 Okay, so the probability that z is less than 4 .930 is, okay, it's around one.

04:07 Okay, it's almost covering all the area.

04:11 Okay, minus the probability that, okay, z less than negative 0 .5862.

04:26 This is 0 .27, okay, 27.

04:33 8 .89.

04:36 Okay.

04:38 So calculating this one and converting 2%, okay.

04:43 So we have 1 minus 2 .7 weekline.

04:51 We have 0 .7 to 11.

04:57 Multiply it by 100%.

04:59 We have 72 .11%...

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