Question

Laptop Computers and Sperm Count Studies have shown that heating the scrotum by just \(1^{\circ} \mathrm{C}\) can reduce sperm count and sperm quality, so men concerned about fertility are cautioned to avoid too much time in the hot tub or sauna. A new study \(^{44}\) suggests that men also keep their laptop computers off their laps. The study measured scrotal temperature in 29 healthy male volunteers as they sat with legs together and a laptop computer on the lap. Temperature increase in the left scrotum over a 60-minute session is given as \(2.31 \pm 0.96\) and a note tells us that "Temperatures are given as \({ }^{\circ} \mathrm{C}\); values are shown as mean \(\pm \mathrm{SD}\)." The abbreviation SD stands for standard deviation. (Men who sit with their legs together without a laptop computer do not show an increase in temperature.) (a) If we assume that the distribution of the temperature increases for the 29 men is symmetric and bell-shaped, find an interval that we expect to contain about \(95 \%\) of the temperature increases. (b) Find and interpret the \(z\) -score for one of the men, who had a temperature increase of \(4.9^{\circ}\).

Short answer

The interval that contains around \(95\%\) of the temperature increases is \(0.39^{\circ}\) to \(4.23^{\circ}\). The \(z-\)score for the man who had a temperature increase of \(4.9^{\circ}\) is 2.7, which means his increase is 2.7 standard deviations above the mean.

Step-by-step solution

Calculation of Interval for Temperature Increase

Given that the mean (standard deviation) temperature increase for the men is \(2.31^{\circ}\pm 0.96^{\circ}\), around \(95\%\) of the observations will lie within two standard deviations of the mean in a normal distribution. Calculate the lower limit of this interval by subtracting two standard deviations from the mean: \(2.31^{\circ} - 2(0.96^{\circ}) = 0.39^{\circ}\), and the upper limit by adding two standard deviations to the mean: \(2.31^{\circ} + 2(0.96^{\circ}) = 4.23^{\circ}\). Therefore, the \(95\%\) confidence interval for the temperature increase is \(0.39^{\circ}\) to \(4.23^{\circ}\).

Calculation of Z-Score for a Particular Temperature Increase

A man whose temperature increase was \(4.9^{\circ}\) is noted. The \(z-\)score for this man can be calculated by subtracting the mean from this value and dividing by the standard deviation: \(z = \frac{4.9^{\circ} - 2.31^{\circ}}{0.96^{\circ}} = 2.7\). This means that this particular man's temperature increase is 2.7 standard deviations above the mean temperature increase.

Key concepts

Standard Deviation

Standard deviation (SD) is a statistical measure that represents the amount of variation or dispersion within a set of values. In simpler terms, it tells us how much the values in a dataset are spread out from the mean (average). If the standard deviation is small, it means the data points are close to the mean, indicating they are similar to each other. Conversely, a larger standard deviation implies greater variability and that the data points are spread out over a wider range of values.

To illustrate, given the problem where the average temperature increase in the scrotum due to a laptop is reported as \(2.31^{\textdegree}C\), with a standard deviation of \(0.96^{\textdegree}C\), we can see that most temperature increases are likely to fall within a range that is close to \(2.31^{\textdegree}C\), with a dispersion within roughly \(0.96^{\textdegree}C\) from that mean. This statistical tool is invaluable in understanding the distribution of data and its inherent variability.

Confidence Interval

A confidence interval gives us a range within which we can be certain, to a certain level of confidence, that a population parameter lies. In most cases, you'll hear about the 95% confidence interval, which is commonly used in the sciences for estimating an unknown parameter. This interval means that if we were to take many samples and construct intervals in the same way, we would expect 95% of them to contain the true parameter.

In the context of the laptop and sperm count study, the 95% confidence interval was calculated to be from \(0.39^{\textdegree}C\) to \(4.23^{\textdegree}C\). This means we are 95% confident that the true mean temperature increase in the scrotum for the overall population, under the same conditions, would fall within this range. It's a way to indicate the precision of our estimate and to account for variability inherent in any sampling process.

Z-Score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It's a way of standardizing scores on the same scale to easily compare different data points. A z-score can indicate how unusual a measurement is; a score of zero represents an average value, while a score of +1 indicates a value that is one standard deviation above the average.

In the exercise, the z-score calculation for the man whose temperature increased by \(4.9^{\textdegree}C\) resulted in a z-score of 2.7. This translates to his temperature increase being 2.7 standard deviations above the mean increase. Such a z-score suggests that this man’s temperature increase is significantly higher than the average, placing him well outside the typical range that would be expected based on the data — a valuable insight indicating an outlier or an extreme value in the context of the study.