To determine the percentage of days with a high temperature below 77 degrees in a city where temperatures are normally distributed with a mean of 69 degrees Fahrenheit and a standard deviation of 6 degrees, we need to follow these steps:
1. Calculate the z-score: The z-score measures how many standard deviations an element is from the mean. The formula for calculating the z-score is:
[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]
where:
- [tex]\( X \)[/tex] is the value for which we are calculating the z-score (77 degrees in this case),
- [tex]\( \mu \)[/tex] is the mean of the distribution (69 degrees),
- [tex]\( \sigma \)[/tex] is the standard deviation of the distribution (6 degrees).
Plugging in the values:
[tex]\[
z = \frac{(77 - 69)}{6} = \frac{8}{6} = 1.3333
\][/tex]
2. Use the z-score to find the cumulative probability: The z-score tells us how far away 77 degrees is from the mean in terms of standard deviations. Next, we use the z-score to find the cumulative probability from the standard normal distribution, which tells us the probability that a value is less than or equal to 77 degrees.
Referring to the z-table (standard normal distribution table) or a cumulative distribution function (CDF) calculator, we find the cumulative probability for a z-score of 1.3333. This cumulative probability represents the area under the normal curve to the left of the z-score.
3. Interpret the cumulative probability: The cumulative probability for a z-score of 1.3333 is approximately 0.9088.
Converting this to a percentage:
[tex]\[
0.9088 \times 100 = 90.88\%
\][/tex]
So, approximately 90.88% of the days have a high temperature below 77 degrees in the vacation resort city.
Among the given options (17.11%, 37.07%, 62.93%, 82.89%), the closest option is not listed. Therefore, it seems there might be a mistake in the provided options or some misinterpretation. The correct answer based on our calculations is 90.88%.
