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\%.
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\%.
