Questions: Melissa is a civil engineer working for the city of Indianapolis on a project to widen some of its downtown streets. One aspect that she has to consider is that, during the hot summer months, the noonday temperature of the asphalt on these streets can get extremely hot. For the project, she models these temperatures of the asphalt using a normal distribution with a mean of 108°F and a standard deviation of 16°F. Use this table or the ALEKS calculator to find the percentage of noonday temperatures of the asphalt between 104°F and 132°F according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23%).

Melissa is a civil engineer working for the city of Indianapolis on a project to widen some of its downtown streets. One aspect that she has to consider is that, during the hot summer months, the noonday temperature of the asphalt on these streets can get extremely hot. For the project, she models these temperatures of the asphalt using a normal distribution with a mean of 108°F and a standard deviation of 16°F.

Use this table or the ALEKS calculator to find the percentage of noonday temperatures of the asphalt between 104°F and 132°F according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23%).

Transcript text: Melissa is a civil engineer working for the city of Indianapolis on a project to widen some of its downtown streets. One aspect that she has to consider is that, during the hot summer months, the noonday temperature of the asphalt on these streets can get extremely hot. For the project, she models these temperatures of the asphalt using a normal distribution with a mean of $108^{\circ} \mathrm{F}$ and a standard deviation of $16^{\circ} \mathrm{F}$. Use this table or the ALEKS calculator to find the percentage of noonday temperatures of the asphalt between $104^{\circ} \mathrm{F}$ and $132{ }^{\circ} \mathrm{F}$ according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example $98.23 \%$ ).

Solution

Solution Steps

Step 1: Define the Normal Distribution Parameters

The temperatures of the asphalt are modeled using a normal distribution with the following parameters:

Mean ($ \mu $): $ 108^{\circ} \mathrm{F} $
Standard Deviation ($ \sigma $): $ 16^{\circ} \mathrm{F} $

Step 2: Calculate the Z-scores

To find the probability that the temperature falls between $ 104^{\circ} \mathrm{F} $ and $ 132^{\circ} \mathrm{F} $, we first calculate the Z-scores for the lower and upper bounds of the range.

The Z-score is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \]

For the lower bound ($ X = 104 $): \[ Z_{start} = \frac{104 - 108}{16} = -0.25 \]

For the upper bound ($ X = 132 $): \[ Z_{end} = \frac{132 - 108}{16} = 1.5 \]

Step 3: Calculate the Probability

Using the Z-scores, we can find the probability that the temperature is between $ 104^{\circ} \mathrm{F} $ and $ 132^{\circ} \mathrm{F} $ using the cumulative distribution function $ \Phi $: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.5) - \Phi(-0.25) \]

From the output, we have: \[ P = 0.5319 \]

Step 4: Convert Probability to Percentage

To express the probability as a percentage: \[ \text{Probability} = P \times 100 = 0.5319 \times 100 = 53.19\% \]