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Suppose that the sitting back-to-knee length for a group of adults has a normal distribution with a mean of [tex]\mu = 23.7 \text{ in.}[/tex] and a standard deviation of [tex]\sigma = 1.2 \text{ in.}[/tex] These data are often used in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. Instead of using 0.05 for identifying significant values, use the criteria that a value [tex]x[/tex] is significantly high if [tex]P(x \text{ or greater}) \leq 0.01[/tex] and a value is significantly low if [tex]P(x \text{ or less}) \leq 0.01[/tex].

Find the back-to-knee lengths separating significant values from those that are not significant.

Back-to-knee lengths greater than [tex]\square[/tex] in. and less than [tex]\square[/tex] in. are not significant, and values outside that range are considered significant.

(Round to one decimal place as needed.)

Is a back-to-knee length of 25.9 in. significantly high?

Sagot :

GinnyAnswer
To answer this question, we need to find the back-to-knee lengths that separate significant values from those that are not significant. In this case, values are deemed significantly high if their probability is less than or equal to 0.01 and significantly low if their probability is similarly low.

Here are the steps:

1. Given Information:
- Mean ([tex]\(\mu\)[/tex]) = 23.7 inches
- Standard deviation ([tex]\(\sigma\)[/tex]) = 1.2 inches
- Significance level for low values = 0.01
- Significance level for high values = 0.01

2. Find the z-scores corresponding to these significance levels:
- For significantly low values, we look at the left tail of the normal distribution curve, where [tex]\( P(X \leq x_{\text{low}}) = 0.01 \)[/tex].
- For significantly high values, we look at the right tail, where [tex]\( P(X \geq x_{\text{high}}) = 0.01 \)[/tex]. This implies that [tex]\( P(X < x_{\text{high}}) = 1 - 0.01 = 0.99 \)[/tex].

3. Convert z-scores to x-values using the mean and standard deviation:
- The z-score corresponding to the 0.01 cumulative probability (left tail) is approximately [tex]\( -2.33 \)[/tex].
- The z-score corresponding to the 0.99 cumulative probability (right tail) is approximately [tex]\( 2.33 \)[/tex].

Using the z-score formula [tex]\( x = \mu + z\sigma \)[/tex]:

a. For significantly low values:
[tex]\[ x_{\text{low}} = 23.7 + (-2.33)(1.2) \approx 23.7 - 2.8 = 20.9 \text{ inches} \][/tex]

b. For significantly high values:
[tex]\[ x_{\text{high}} = 23.7 + (2.33)(1.2) \approx 23.7 + 2.8 = 26.5 \text{ inches} \][/tex]

4. Determining if a length of 25.9 inches is significantly high:
- We compare 25.9 inches to the critical value for significantly high lengths ([tex]\(x_{\text{high}} = 26.5 \text{ inches}\)[/tex]).
- Since 25.9 inches is less than 26.5 inches, it is not considered significantly high.

Thus, the back-to-knee lengths separating significant values from those that are not significant are:
- Significantly low values: Less than 20.9 inches.
- Significantly high values: Greater than 26.5 inches.

In conclusion:
- Back-to-knee lengths greater than 20.9 inches and less than 26.5 inches are not significant.
- Values outside that range (i.e., less than 20.9 inches or greater than 26.5 inches) are considered significant.

Therefore, a back-to-knee length of 25.9 inches is not significantly high.
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