Mean minus variance after converting to Celsius

Question

Daily June highs in a city average $80^{\circ}$ F with a standard deviation of $5^{\circ}$ F. Convert to Celsius and report $|\text{mean}-\text{variance}|$ of the Celsius temperature, rounded to two decimals.

Short answer

$18.95$ . With $C=\tfrac59(F-32)$ , the mean is $\tfrac59\cdot48\approx26.67$ and the variance is $(\tfrac59)^2\cdot25\approx7.72$ (the shift does not affect variance), giving $26.67-7.72=18.95$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Covariance and Correlation
Populations, Samples, and Sampling Distributions

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