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Too cool at the cabin? During the winter months, the temperatures at the Starneses' Colorado cabin can stay well below freezing \(\left(32^{\circ} \mathrm{F} \text { or } 0^{\circ} \mathrm{C}\right)\) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at \(50^{\circ} \mathrm{F}\) . She also buys a digital thermometer that records the indoor temperature each night at midnight. Unfortunately, the thermometer is programmed to measure the temperature in degrees Celsius. Based on several years' worth of data, the temperature \(T\) in the cabin at midnight on a randomly selected night follows a Normal distribution with mean \(8.5^{\circ} \mathrm{C}\) and standard deviation \(2.25^{\circ} \mathrm{C} .\) (a) Let \(Y=\) the temperature in the cabin at midnight on a randomly selected night in degrees Fahrenheit (recall that \(\mathrm{F}=(9 / 5) \mathrm{C}+32 ) .\) Find the mean and standard deviation of \(\mathrm{Y}\) . (b) Find the probability that the midnight temperature in the cabin is below \(40^{\circ} \mathrm{F}\) . Show your work.

Step by step solution

01

Convert Mean Temperature to Fahrenheit

We know that the mean temperature in Celsius is \(\mu_C = 8.5^{\circ} C\). To convert this to Fahrenheit, use the formula: \[ F = \left(\frac{9}{5}\right)C + 32 \] So, the mean in Fahrenheit, \(\mu_F\), is: \[ \mu_F = \left(\frac{9}{5}\right)\times 8.5 + 32 = 47.3^{\circ} F \]

02

Convert Standard Deviation to Fahrenheit

The standard deviation in Celsius is \(\sigma_C = 2.25^{\circ} C\). The conversion to Fahrenheit involves only the multiplicative factor since it doesn't involve shifting the origin (the 32 degrees), so: \[ \sigma_F = \left(\frac{9}{5}\right) \times 2.25 \approx 4.05^{\circ} F \]

03

Probability Conversion to Z-score

Now, to find the probability that the temperature is below \(40^{\circ} F\), we first convert \(40^{\circ} F\) to a Z-score. The Z-score formula is: \[ Z = \frac{Y - \mu_F}{\sigma_F} \]Where \(Y = 40^{\circ} F\). Thus,\[ Z = \frac{40 - 47.3}{4.05} \approx -1.80 \]

04

Find Probability Using Z-table

To find the probability that the temperature is below \(40^{\circ} F\), we look up the Z-score of \(-1.80\) in the standard normal distribution table. A Z-score of \(-1.80\) corresponds to a probability of approximately \(0.0359\), or \(3.59\%\). Therefore, there is a 3.59% chance the temperature is below \(40^{\circ} F\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Conversion

Temperature conversion is an important skill, especially when dealing with data in different units like Celsius and Fahrenheit. In this exercise, we need to convert temperatures between these two units using the formula:

• Celsius to Fahrenheit: \( F = \left(\frac{9}{5}\right)C + 32 \)
• Fahrenheit to Celsius: \( C = \left(\frac{5}{9}\right)(F - 32) \)

To convert a mean temperature of \(8.5^{\circ} \text{C} \) to Fahrenheit, we plug \( C = 8.5 \) into the formula and find that the mean temperature is \( 47.3^{\circ} \text{F} \).
Remember that when converting standard deviation, only the multiplier \(\frac{9}{5}\) is used, since the intercept \(32\) does not affect variability.
This results in a standard deviation of \(4.05^{\circ} \text{F} \) when the original standard deviation is \(2.25^{\circ} \text{C} \). Understanding these conversions helps ensure accuracy when switching between systems, crucial for analyzing temperature data correctly.

Mean and Standard Deviation

Mean and standard deviation are fundamental concepts in statistics that describe a data set's central tendency and spread, respectively.
The mean temperature, \(\mu\), indicates the average temperature over time. For Celsius, it's given as \(8.5^{\circ}\) in this exercise. After converting it to Fahrenheit for consistency, we get \(47.3^{\circ} F\).
Standard deviation, \(\sigma\), measures the variation or dispersion of the data points from the mean. A smaller standard deviation means the data points are closer to the mean, while a larger one indicates more spread out data.

• In Celsius: \(\sigma_C = 2.25^{\circ} C\)
• Converted to Fahrenheit: \(\sigma_F = 4.05^{\circ} F\)

These values help us understand not just the average temperature in the cabin but also the extent to which temperatures can vary from this average, providing a comprehensive view of the cabin's climate profile.

Probability Calculations

Probability calculations are valuable for determining the likelihood of a specific event happening within a normal distribution. With the cabin's temperature data, we calculate the probability that the temperature falls below \(40^{\circ} F\).
First, the Z-score is calculated using the formula: \[ Z = \frac{Y - \mu_F}{\sigma_F} \]where \(Y = 40^{\circ} F\), \(\mu_F = 47.3^{\circ} F\), and \(\sigma_F = 4.05^{\circ} F\). This yields a Z-score of approximately \(-1.80\), indicating how many standard deviations \(40^{\circ} F\) is below the mean.
Once we have the Z-score, we use the standard normal distribution table to find the probability. A Z-score of \(-1.80\) corresponds to a probability of \(0.0359\), or \(3.59\%\).
This means there's a \(3.59\%\) chance that the indoor temperature drops below \(40^{\circ} F\), reflecting the frequency of particularly cold nights in the cabin.