Question

The mean of the 100 car speeds in Exercise 20 was \(23.84 \mathrm{mph}\), with a standard deviation of \(3.56 \mathrm{mph}\). a) Using a Normal model, what values should border the middle \(95 \%\) of all car speeds? b) Here are some summary statistics. $$ \begin{array}{lll} \hline \text { Percentile } & & \text { Speed } \\ \hline 100 \% & \text { Max } & 34.060 \\ 97.5 \% & & 30.976 \\ 90.0 \% & & 28.978 \\ 75.0 \% & \text { Q3 } & 25.785 \\ 50.0 \% & \text { Median } & 23.525 \\ 25.0 \% & \text { Q1 } & 21.547 \\ 10.0 \% & & 19.163 \\ 2.5 \% & & 16.638 \\ 0.0 \% & \text { Min } & 16.270 \\ \hline \end{array} $$

Short answer

The middle 95% of car speeds range from approximately 16.862 mph to 30.818 mph.

Step-by-step solution

Understanding the Normal Distribution

To find the values that border the middle 95% of car speeds, we use the properties of the normal distribution. In a normal distribution, the middle 95% is captured within approximately two standard deviations from the mean.

Calculating Z-scores for 95% Middle Range

For the middle 95% of a normal distribution, the z-scores corresponding to the lower and upper bounds are approximately -1.96 and 1.96, based on standard normal distribution tables.

Calculate Lower Bound Value

To find the lower boundary value, use the formula: \( \text{Lower Bound} = \mu + (z \times \sigma) \), where \( \mu = 23.84 \) and \( z = -1.96 \) and \( \sigma = 3.56 \). Substitute these into the equation:\[ \text{Lower Bound} = 23.84 + (-1.96 \times 3.56) = 23.84 - 6.978 = 16.862 \]

Calculate Upper Bound Value

To find the upper boundary value, similarly use: \( \text{Upper Bound} = \mu + (z \times \sigma) \), where \( \mu = 23.84 \), \( z = 1.96 \) and \( \sigma = 3.56 \). Substitute these into the equation:\[ \text{Upper Bound} = 23.84 + (1.96 \times 3.56) = 23.84 + 6.978 = 30.818 \]

Confirm with Given Summary Statistics

The summary statistics provided show that the 2.5th percentile is at a speed of 16.638, and the 97.5th percentile is at a speed of 30.976. These match our calculated bounds of the middle 95%, confirming the calculations.

Key concepts

Z-scores

The concept of Z-scores is crucial in understanding how data points relate to the mean of a dataset. A Z-score measures how many standard deviations an element is from the mean. For example, in a normal distribution, a Z-score of 0 means the element is exactly at the mean. A positive Z-score indicates the data point is above the mean, while a negative Z-score signifies it is below the mean.
To find Z-scores, you use the formula: \[Z = \frac{(X - \mu)}{\sigma}\]where \(X\) is a data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
This measure is particularly useful when comparing different datasets or when trying to find the probability of a result occurring within a certain part of a normal distribution. For instance, when determining the range of the middle 95% of any dataset, the Z-scores are usually -1.96 and 1.96.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates the data points tend to be close to the mean of the set, while a high standard deviation indicates the data points are spread out over a wider range of values.
In mathematical terms, the formula for standard deviation \(\sigma\) is:\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2}\]where \(N\) is the number of data points, \(X_i\) represents each data point, and \(\mu\) is the mean of the dataset.
Standard deviation is a critical component of the normal distribution. It helps to determine the confidence intervals and probability of values occurring within that range. It is frequently used in various fields like finance, physics, and social sciences to compare different datasets by providing a scale that measures the variability of data.

Mean

The mean, also known as the average, is the sum of all numbers in a dataset divided by the number of numbers in that dataset. This is one of the most straightforward and commonly used statistical measures.
The formula to calculate mean \(\mu\) is:\[\mu = \frac{1}{N} \sum_{i=1}^{N} X_i\]where \(N\) is the total number of values, and \(X_i\) represents each value in the dataset.
In the context of a normal distribution, the mean is the central point around which data is distributed, forming the peak of the bell curve. This is why the mean is also regarded as a measure of central tendency. It's important to note that the mean can be influenced by outliers or extreme values. Therefore, in datasets with significant skewness, other measures like the median might provide better insight.