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------------------------------------------------ Select the correct answer.

The average summer temperature in Anchorage is [tex]$69^{\circ} F$[/tex]. If the daily temperature is normally distributed with a standard deviation of [tex]$7^{\circ} F$[/tex], what percentage of the time would the temperature be between [tex]$55^{\circ} F$[/tex] and [tex]$76^{\circ} F$[/tex]?

A. [tex]$34.0 \%$[/tex]
B. [tex]$47.5 \%$[/tex]
C. [tex]$68.0 \%$[/tex]
D. [tex]$81.5 \%$[/tex]
E. [tex]$99.7 \%$[/tex]

Answer :

To determine the percentage of time the temperature is between [tex]\(55^\circ F\)[/tex] and [tex]\(76^\circ F\)[/tex], we need to understand how data behaves under a normal distribution given the mean and standard deviation.

Here's a step-by-step explanation:

1. Identify the Given Information:
- The average (mean) temperature is [tex]\(69^\circ F\)[/tex].
- The standard deviation is [tex]\(7^\circ F\)[/tex].
- We are looking for the percentage of time the temperature is between [tex]\(55^\circ F\)[/tex] and [tex]\(76^\circ F\)[/tex].

2. Calculate the Z-Scores:
- The Z-score allows us to determine how many standard deviations a particular value is from the mean.
- For the lower bound ([tex]\(55^\circ F\)[/tex]):
[tex]\[
z_{lower} = \frac{55 - 69}{7} = -2.0
\][/tex]
- For the upper bound ([tex]\(76^\circ F\)[/tex]):
[tex]\[
z_{upper} = \frac{76 - 69}{7} = 1.0
\][/tex]

3. Use the Z-Scores to Find Probabilities:
- Using a standard normal distribution table, or a calculator, find the cumulative probability for each Z-score.
- The cumulative probability for [tex]\(z_{lower} = -2.0\)[/tex] and [tex]\(z_{upper} = 1.0\)[/tex] gives us the likelihood of the temperature being less than the specified Z-scores.

4. Find the Difference Between Probabilities:
- The probability that the temperature is between [tex]\(55^\circ F\)[/tex] and [tex]\(76^\circ F\)[/tex] is the difference between the cumulative probability of [tex]\(z_{upper}\)[/tex] and [tex]\(z_{lower}\)[/tex].

5. Convert Probability to Percentage:
- Once you have calculated the probability, convert it to a percentage for easier interpretation.
- In this scenario, the calculated probability that the temperature falls within this range is approximately [tex]\(81.86\%\)[/tex].

Thus, about [tex]\(81.5\%\)[/tex] of the time, the daily temperature in Anchorage is expected to be between [tex]\(55^\circ F\)[/tex] and [tex]\(76^\circ F\)[/tex]. Therefore, the correct answer is D. 81.5\%.