91Ó°ÊÓ

When dealing with statistics, the concept of a z-score is fundamental. It indicates how many standard deviations an element is from the mean, thus providing a way to compare different data points. Considering normal distribution, such as the temperatures in New York City, the formula for the z-score is \[ z = \frac{x-\mu}{\sigma} \] where \( x \) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

For example, if we want to determine how unusual the freezing point of \(32^\circ \mathrm{F}\) is compared to the mean February temperature, we use the mean \(27^\circ \mathrm{F}\) and standard deviation \(6^\circ \mathrm{F}\) to find a z-score. In the context of this exercise, the freezing point temperature's z-score is about 0.833, suggesting it is less than one standard deviation above the mean February temperature.

Standard deviation is a critical statistical measure that tells us how spread out the numbers are in a data set. It is the square root of the variance, which is the average of the squared differences from the mean.

In practical terms, in our New York City temperature example, a standard deviation of \(6^\circ \mathrm{F}\) means that the majority of temperature values lie within a range of \(6^\circ \mathrm{F}\) above or below the mean (27°F). A smaller standard deviation would suggest that the daily temperatures are clustered more closely around the mean, while a larger one would indicate more variability. Thus, knowing the standard deviation helps us understand the variability in February's minimum temperature.

Probability is a way of expressing the likelihood of an event occurring, often represented as a number between 0 and 1. In the context of normal distribution, probabilities are associated with areas under the curve.

To find the probability of a temperature being below freezing, we use the z-score that corresponds to \(32^\circ \mathrm{F}\). Then, with the help of a z-table or a calculator, we can determine that approximately 79.7% of the data (or days) are warmer than freezing. Since we're interested in days colder than freezing, we need to look at the lower tail of the distribution. By doing so, we find that about 20.3% of the days are expected to have minimum temperatures below freezing, indicating the probability of such an event.

Normal distribution temperature refers to how the daily temperatures are distributed in a bell-curve pattern around the mean temperature. It assumes that most days will have temperatures close to the mean, with fewer days showing extreme cold or warm temperatures.

In our scenario, the minimum temperatures in February for New York City are said to be normally distributed with a mean of \(27^\circ \mathrm{F}\) and a standard deviation of \(6^\circ \mathrm{F}\). This normal distribution allows us to make predictions, like the calculation of the proportion of days with temperatures below freezing. Therefore, understanding the principles of normal distribution helps to anticipate weather patterns and temperature trends for any given month.