Binomial probability #4 - Questions and Answers

Question 1 A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his southern Florida home last year. The coldest temperature for any week was 36°F, but he inadvertently recorded the Celsius value of 2°. Assuming that he correctly listed all the other temperatures, explain how this error will affect these summary statistics: a) measures of center: mean and median. b) measures of spread: range, IQR, and standard deviation. Answer 1 a) The median will remain unchanged, while the mean will decrease. Because the lowest value decreased and the median is not influenced by one outlier while the mean is influenced by it. b) The range and standard deviation will increase, while the IQR remains unchanged. Because the IQR is unaffected by an outlier, while the range and standard deviation are influenced by it. Result: a) Median unchanged, mean decreases b) Range and standard deviation increase, IQR unchanged Answer 2 a) The error in recording the coldest temperature in Celsius instead of Fahrenheit will affect the measures of center, specifically the mean and median. For the mean, the incorrect Celsius value of 2°C needs to be converted to Fahrenheit to match the other temperatures. The conversion formula from Celsius to Fahrenheit is given by 𝐹 = 9 5 𝐶 + 32 F= 5 9 ​ C+32. Applying this formula, the incorrect Celsius value of 2°C is equivalent to 𝐹 = 9 5 ( 2 ) + 32 = 35.6 ° 𝐹 F= 5 9 ​ (2)+32=35.6°F. Therefore, the correct coldest temperature is 35.6°F instead of 36°F. This adjustment will slightly decrease the mean since the original value of 36°F is higher than the correct value of 35.6°F. The effect on the median will depend on the distribution of the remaining temperatures. b) The error will affect the measures of spread, including the range, interquartile range (IQR), and standard deviation. The range is the difference between the maximum and minimum values. Since the coldest temperature was incorrectly recorded as 36°F instead of the correct value of 35.6°F, the range will be slightly higher than what it should have been. The interquartile range (IQR) is a measure of the spread of the middle 50% of the data. The error in recording the coldest temperature does not directly affect the IQR since it focuses on the central part of the data, but it may indirectly influence it if the coldest temperature falls within the lower quartile or upper quartile. The standard deviation measures the dispersion of the data from the mean. Since the mean will be slightly lower due to this adjustment, the standard deviation may also be reduced. However, the magnitude of this effect will depend on the distribution and variability of the other temperatures. Overall conclusion: The error in recording the temperature in Celsius instead of Fahrenheit will have minor effects on the measures of center and spread, with potential impacts on the mean, range, and standard deviation. Answer 3 To understand how the error in recording the temperature will affect the summary statistics, let’s analyze each statistic individually. a) Measures of Center: Mean ( 𝑥 ˉ x ˉ ): The mean is the average of all the temperatures. Converting the recorded temperature of 2°C to Fahrenheit reveals that it should have been approximately 35.6°F. Including this corrected value in the dataset will slightly decrease the mean, as the corrected value is lower than the coldest recorded temperature (36°F). Median ( 𝑥 x): The median is the middle value when the temperatures are arranged in ascending order. Since the corrected value of 35.6°F falls within the recorded temperatures, it will not affect the median. The median will remain the same. b) Measures of Spread: Range: The range is the difference between the highest and lowest values. As the coldest recorded temperature remains the same at 36°F, the range will not be affected by the error. Interquartile Range (IQR): The IQR measures the range of the middle 50% of the temperatures. Since the coldest recorded temperature remains the same, the error will not impact the IQR. Standard Deviation ( 𝜎 σ): The standard deviation measures the dispersion of the temperatures around the mean. The error will affect the standard deviation because the corrected value of 35.6°F is slightly lower than the coldest recorded temperature of 36°F. The corrected value being lower will result in a slightly smaller standard deviation. In summary, the error in recording the temperature will have the following effects on the summary statistics: a) The mean will decrease slightly. b) The median, range, and IQR will remain the same. c) The standard deviation will decrease slightly. Answer 4 The error in recording the Celsius value instead of the Fahrenheit value for the coldest temperature will affect the mean, range, and standard deviation, while the median and IQR will remain unaffected. Explanation: The error in recording the Celsius value instead of the Fahrenheit value for the coldest temperature will affect the measures of center, specifically the mean and median. Let’s denote the set of weekly low temperatures in degrees Fahrenheit as 𝑋 X and the set of temperatures with the Celsius value of 2 degrees as 𝑌 Y. Since the meteorologist inadvertently recorded 2 ° 𝐶 2°C as the lowest temperature, we can represent this error as 𝑌 = ( 2 ) Y=(2). The correct set of temperatures without this error would be 𝑋 − 𝑌 X−Y. The mean is calculated by summing up all the values and dividing by the total count. The mean of the temperatures with the error included would be: Mean with error = Σ 𝑋 + 2 𝑛 Mean with error ​ = n ΣX+2 ​ where 𝑛 n is the total count of temperatures in 𝑋 X. However, to find the mean without the error, we need to exclude the value of 2 ° 𝐹 2°F from the sum. Therefore, the mean without the error would be: Mean without error = Σ ( 𝑋 − 𝑌 ) 𝑛 Mean without error ​ = n Σ(X−Y) ​ The error will affect the mean by increasing it due to the inclusion of the additional value of 2 ° 𝐹 2°F in the calculation. Median: The median is the middle value when the data is arranged in ascending order. The median will not be affected by the error in this case since the value of 2 ° 𝐹 2°F is the lowest temperature and would not be the median value. b) Measures of Spread: The error in recording the Celsius value instead of the Fahrenheit value for the coldest temperature will also affect the measures of spread, specifically the range, interquartile range (IQR), and standard deviation. Range: The range is the difference between the maximum and minimum values. The maximum value will not be affected by the error, but the minimum value will change due to the inclusion of 2 ° 𝐹 2°F. Therefore, the range will increase as a result of the error. Interquartile Range (IQR): The IQR is the range of the middle 50% of the data, calculated as the difference between the first quartile (Q1) and the third quartile (Q3). Since the error only affects the minimum value and not the quartiles, the IQR will not be directly affected by the error. Standard Deviation: The standard deviation measures the spread of the data around the mean. Since the mean is affected by the error, the standard deviation will also be influenced. Including the value of 2 ° 𝐹 2°F in the dataset will increase the variability, leading to a potentially higher standard deviation.