It takes the average adult 22 minutes to walk a mile, with a standard deviation of 6 minutes. About what percentage of adults take longer than 27 minutes to walk a mile?

Note: Assume that a normal model is appropriate for the distribution of walking times.

\begin{tabular}{|r|l|l|l|l|l|l|l|l|}
\hline \multicolumn{8}{|c|}{Table shows values to the LEFT of the [tex]$z$[/tex]-score} \\
\hline[tex]$z$[/tex] & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 & 0.07 & 0.08 & 0.09 \\
\hline 0.8 & 0.79389 & 0.79673 & 0.79955 & 0.80234 & 0.80511 & 0.80785 & 0.81057 & 0.81327 \\
\hline 0.9 & 0.82121 & 0.82381 & 0.82639 & 0.82894 & 0.83147 & 0.83398 & 0.83646 & 0.83891 \\
\hline 1.0 & 0.84614 & 0.84849 & 0.85083 & 0.85314 & 0.85543 & 0.85769 & 0.85993 & 0.86214 \\
\hline 1.1 & 0.86864 & 0.87076 & 0.87286 & 0.87493 & 0.87698 & 0.87900 & 0.88100 & 0.88298 \\
\hline 1.2 & 0.88877 & 0.89065 & 0.89251 & 0.89435 & 0.89617 & 0.89796 & 0.89973 & 0.90147 \\
\hline-1.2 & 0.11123 & 0.10935 & 0.10749 & 0.10565 & 0.10383 & 0.10204 & 0.10027 & 0.09853 \\
\hline-1.1 & 0.13136 & 0.12924 & 0.12714 & 0.12507 & 0.12302 & 0.12100 & 0.11900 & 0.11702 \\
\hline-1.0 & 0.15386 & 0.15151 & 0.14917 & 0.14686 & 0.14457 & 0.14231 & 0.14007 & 0.13786 \\
\hline-0.9 & 0.17879 & 0.17619 & 0.17361 & 0.17106 & 0.16853 & 0.16602 & 0.16354 & 0.16109 \\
\hline-0.8 & 0.20611 & 0.20327 & 0.20045 & 0.19766 & 0.19489 & 0.19215 & 0.18943 & 0.18673 \\
\hline
\end{tabular}

A. 29.67%
B. 79.67%
C. 20.33%



Answer :

To determine what percentage of adults take longer than 27 minutes to walk a mile, we will follow these steps:

### Step 1: Understand the Parameters
1. The average (mean) walking time is 22 minutes.
2. The standard deviation of the walking times is 6 minutes.
3. We want to find the probability of an individual taking longer than 27 minutes.

### Step 2: Calculate the Z-Score
First, we need the z-score which standardizes the value. The z-score indicates how many standard deviations a data point (walking time of 27 minutes) is from the mean.

The formula for the z-score [tex]\( z \)[/tex] is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value we are looking at (27 minutes)
- [tex]\( \mu \)[/tex] is the mean (22 minutes)
- [tex]\( \sigma \)[/tex] is the standard deviation (6 minutes)

Substituting the values in:
[tex]\[ z = \frac{27 - 22}{6} \][/tex]
[tex]\[ z = \frac{5}{6} \][/tex]
[tex]\[ z \approx 0.833 \][/tex]

### Step 3: Use the Z-Score to Find the Cumulative Probability
Using the z-score, we can determine the cumulative probability from the standard normal distribution table.

The z-score we found is 0.833. Referring to the closest values in the provided table:
- For [tex]\( z = 0.83 \)[/tex], it shows the probability to the left of this z-score is approximately 0.7967.

### Step 4: Find the Probability of Taking Longer than 27 Minutes
The cumulative probability provides the proportion of the data to the left of the z-score. To find the percentage of adults who take longer than 27 minutes, we must subtract this probability from 1 (since the total probability is 1).

[tex]\[ P(X > 27) = 1 - P(X \le 27) = 1 - 0.7967 = 0.2033 \][/tex]

### Step 5: Convert the Probability to a Percentage
Finally, convert the probability to a percentage:
[tex]\[ 0.2033 \times 100 \approx 20.33\% \][/tex]

### Conclusion
Approximately [tex]\( 20.33\% \)[/tex] of adults take longer than 27 minutes to walk a mile.

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