Questions: The average winter daily temperature in a particular city has a distribution that is approximately Normal, with a mean of 28.2°F and a standard deviation of 7.9°F. What percentage of winter days in this city have a daily temperature of 35°F or warmer? The percentage of winter days in this city that have a daily temperature of 35°F or warmer is %.

Solution Steps

To find the Z-score for a daily temperature of \( 35.0^{\circ} \mathrm{F} \), we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

• \( X = 35.0 \)
• \( \mu = 28.2 \)
• \( \sigma = 7.9 \)

Substituting the values, we get:

\[ z = \frac{35.0 - 28.2}{7.9} = 0.8608 \]

Next, we need to find the probability that the temperature is \( 35^{\circ} \mathrm{F} \) or warmer. This is represented as:

\[ P(X \geq 35) = P(Z \geq 0.8608) = 1 - P(Z < 0.8608) \]

Using the cumulative distribution function \( \Phi \), we have:

\[ P(Z \geq 0.8608) = 1 - \Phi(0.8608) \]

From the standard normal distribution table or calculator, we find:

\[ \Phi(0.8608) \approx 0.1947 \]

Thus,

\[ P(Z \geq 0.8608) = 1 - 0.1947 = 0.8053 \]

To express this probability as a percentage, we multiply by 100:

\[ \text{Percentage} = 0.8053 \times 100 = 80.53\% \]

However, since we are interested in the percentage of winter days with a temperature of \( 35^{\circ} \mathrm{F} \) or warmer, we take the complement of the probability:

\[ \text{Percentage of days} = 19.47\% \]

Final Answer