Question

A town's January high temperatures average \(36^{\circ} \mathrm{F}\) with a standard deviation of \(10^{\circ}\), while in July the mean high temperature is \(74^{\circ}\) and the standard deviation is \(8^{\circ} .\) In which month is it more unusual to have a day with a high temperature of \(55^{\circ} ?\) Explain.

Short answer

A high of \(55^{\circ}\) is more unusual in July because the Z-score (-2.375) is higher in magnitude than in January (1.9).

Step-by-step solution

Find the Z-score for January

To determine how unusual a day with a high temperature of \(55^{\circ}\) is in January, compute the Z-score using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \(X = 55^{\circ}\), \(\mu = 36^{\circ}\) (mean for January), and \(\sigma = 10^{\circ}\) (standard deviation for January). Plugging in these values, we get:\[ Z = \frac{55 - 36}{10} = \frac{19}{10} = 1.9 \]

Find the Z-score for July

Similarly, compute the Z-score for July using the same formula:\[ Z = \frac{X - \mu}{\sigma} \]where \(X = 55^{\circ}\), \(\mu = 74^{\circ}\) (mean for July), and \(\sigma = 8^{\circ}\) (standard deviation for July).Plugging in these values, we get:\[ Z = \frac{55 - 74}{8} = \frac{-19}{8} = -2.375 \]

Compare Z-scores

Z-score indicates how many standard deviations an element is from the mean. A higher absolute Z-score means a more unusual event. For January, the Z-score is 1.9, and for July, it is -2.375. Since the absolute value of the July Z-score (2.375) is greater than the value for January (1.9), a \(55^{\circ}\) day is more unusual in July.

Key concepts

Standard Deviation

Standard deviation is a measure used in statistics to quantify the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the numbers in a data set are. If the standard deviation is low, it means most numbers are close to the average value (mean) of the set. On the other hand, a high standard deviation means numbers are more spread out from the mean.

• This measure is crucial because it helps us understand the consistency of data.
• A large standard deviation indicates greater variability in high temperatures, while a smaller one suggests more consistency.

Think of it like this: in a classroom where every student scores close to the average, the standard deviation is low. But if some students score very high and some score very low, the standard deviation is high.

Mean

The mean, also known as the average, is a central concept in mathematics and statistics. It is calculated by adding up all the values in a data set and then dividing by the number of values in that set. In our weather example, if you want to find the mean high temperature for January, you would sum up all the high temperatures for the days in January and divide by the number of days.

• The mean provides a simple yet powerful way to summarize a large amount of data with a single number.
• It gives us a center point around which we can explore variations, such as through the use of the standard deviation.

Understanding the mean helps in expressing what is typical in a data set, like the usual high temperature in a given month.

High Temperatures

High temperatures are simply the peak temperatures recorded over a specific period, like daily maximum temperatures in a month. For towns and cities, these high temperatures are significant as they affect daily life and activities, such as agriculture and energy consumption. An unusual high temperature is generally one that significantly deviates from what is expected or average for that time.

• By looking at high temperatures, people often assess weather patterns and anomalies.
• Weather data often distinguishes between temperatures recorded during the day and night, focusing usually on the highest (max temperature) and lowest (min temperature).

In the town's January and July temperature data, the focus is on high temperatures, which indicate the most extreme points in daily temperature fluctuations.

Normal Distribution

Normal distribution is a concept in statistics that describes how data points are typically spread across a range of values. Often referred to as a "bell curve," because of its shape, normal distribution assumes that data near the mean is more frequent in occurrence than data far from the mean. This distribution is symmetric, with most values clustering around a central region.

• It is key in understanding how unusual a certain data point is compared to the rest.
• Normal distribution curves are crucial for calculating probabilities and determining Z-scores, which help in assessing how typical or atypical a particular observation is.

In our example of high temperatures, figuring out whether a specific day is unusual involves understanding how that temperature fits within the normal distribution of temperatures over that month.