Cold cabin? During the winter months, the temperatures at the Colorado cabin owned by the Starnes family can stay well below freezing (32°F or 0°C) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50°F. The manufacturer claims that the thermostat allows variation in home temperature of s = 3°F. Mrs. Starnes suspects that the manufacturer is overstating how well the thermostat works.
The correct answer and explanation is:
Answer:
Mrs. Starnes is concerned that the thermostat in their Colorado cabin is not maintaining the promised temperature variation of s=3∘Fs = 3^\circ F. To verify this, she could conduct a hypothesis test:
- Null Hypothesis (H0H_0): The standard deviation of the temperature is σ=3∘F\sigma = 3^\circ F, as the manufacturer claims.
- Alternative Hypothesis (HaH_a): The standard deviation of the temperature is greater than 3°F, meaning the thermostat is less accurate than claimed.
To test this, Mrs. Starnes should record temperature readings over time and compute the sample standard deviation ss. If the sample standard deviation is significantly higher than 3°F, she has statistical evidence to doubt the manufacturer’s claim.
Explanation:
In colder climates, maintaining stable indoor temperatures is crucial to prevent frozen pipes and structural damage. A thermostat is designed to regulate temperature within a set range, but variations in performance can cause deviations from the expected values.
The standard deviation measures the extent to which the actual temperatures fluctuate from the average set temperature (50°F). If the manufacturer’s claim (σ=3∘F\sigma = 3^\circ F) is correct, the variation in recorded temperatures should remain within that limit. However, if Mrs. Starnes records a larger standard deviation over time, it indicates that the thermostat might not be controlling the temperature as precisely as advertised.
By conducting a statistical test (such as a chi-square test for variance), she can determine whether the observed standard deviation significantly exceeds 3°F. If so, she could consider recalibrating the thermostat, getting a replacement, or using an additional heat source to ensure stability.
Image Generation:
I will now generate an image illustrating temperature fluctuations inside the cabin over time, showing how the actual temperatures compare to the expected 50°F ± 3°F range.
Here’s the generated image depicting the cozy cabin in Colorado with temperature fluctuations beyond the expected range. Let me know if you need any adjustments or further analysis!
