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The Thundering Herd, an amusement park ride, is not open to patrons less than [tex]54[/tex] inches tall. If the mean height of park patrons is [tex]68[/tex] inches with a standard deviation of 12 inches, what percent of the patrons will not be able to use this ride?

1. Calculate the [tex]z[/tex]-score for [tex]54[/tex] inches:
[tex]z[/tex] for [tex]54[/tex] inches = [tex]\square[/tex]
(Note: The negative means 54" is less than the mean of 68". Do not enter a negative percent.)

2. The percentage for the above [tex]z[/tex] is [tex]\square[/tex]%.

3. This is the percentage of patrons between [tex]54[/tex] inches and [tex]68[/tex] inches.

4. The percentage for all patrons above [tex]68[/tex] inches is [tex]\square[/tex]%.
(This corresponds to [tex]z = +4[/tex].)

5. So the percentage of patrons above [tex]54[/tex] inches is [tex]\square[/tex]%.

6. Therefore, adding the two together, the percentage of patrons below 54" and who may not use this ride is [tex]\square[/tex]%.

Answer :

To determine the percentage of patrons who will not be able to use the amusement park ride, we can follow a step-by-step approach using statistical concepts. Let's break it down:

1. Understanding the Question:
- The ride has a height requirement: patrons must be at least 54 inches tall.
- The average (mean) height of patrons is 68 inches, with a standard deviation of 12 inches.
- We want to find out what percentage of patrons are below 54 inches in height.

2. Calculate the Z-score:
- A Z-score is a measure of how far a data point is from the mean, in terms of standard deviations.
- Z-score formula: [tex]\( z = \frac{(X - \mu)}{\sigma} \)[/tex]
- [tex]\( X \)[/tex] is the value of interest (54 inches here).
- [tex]\( \mu \)[/tex] is the mean (68 inches).
- [tex]\( \sigma \)[/tex] is the standard deviation (12 inches).
- Substitute the values:
[tex]\[
z = \frac{(54 - 68)}{12} = -1.167
\][/tex]

3. Find the Percentage Below 54 Inches:
- The Z-score tells us how many standard deviations away 54 inches is from the mean.
- Using a standard normal distribution table, or cumulative distribution function (CDF), we can find the area to the left of this Z-score. This area represents the percentage of patrons shorter than 54 inches.
- For [tex]\( z = -1.167 \)[/tex], the percentage of patrons below 54 inches is approximately 12.17%.

4. Interpret Additional Information:
- The question mentions percentages for patrons both below and above the mean, but the focus is primarily on those below 54 inches.
- Since we're mainly interested in who cannot ride (those below 54 inches), we summarize this as:

5. Conclusion:
- 12.17% of the patrons will not be able to use the ride because they are shorter than 54 inches.

In summary, about 12.17% of park patrons will be too short to go on "The Thundering Herd" ride based on the height requirement.

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