Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars:
The standard deviation of X is sX = 1.08. If many households were selected at random, which of the following would be the best interpretation of the value 1.08?
(a) The mean number of cars would be about 1.08.
(b) The number of cars would typically be about 1.08 from the mean.
(c) The number of cars would be at most 1.08 from the mean.
(d) The number of cars would be within 1.08 from the mean about 68% of the time.
(e) The mean number of cars would be about 1.08 from the expected value.
The Correct Answer and Explanation is :
The correct answer is:
(d) The number of cars would be within 1.08 from the mean about 68% of the time.
Explanation:
The standard deviation, ( s_X = 1.08 ), measures how spread out the values of the random variable ( X ) (the number of cars owned by a household) are around the mean. This is a key concept in probability and statistics.
The standard deviation is a measure of variability or dispersion, which tells us how much individual values deviate from the mean. In the context of a random variable, this means that most of the observed values of ( X ) (the number of cars) are likely to fall within a certain range of the mean. For a normal distribution (or approximately normal distribution, if we assume that the distribution of ( X ) is roughly symmetric), about 68% of the values of ( X ) will fall within one standard deviation from the mean. This is a well-known rule from statistics called the 68-95-99.7 rule (also called the empirical rule), which states:
- 68% of the data will fall within one standard deviation from the mean.
- 95% of the data will fall within two standard deviations from the mean.
- 99.7% of the data will fall within three standard deviations from the mean.
In this case, the standard deviation is 1.08, so about 68% of households would own between ( \mu – 1.08 ) and ( \mu + 1.08 ) cars, where ( \mu ) represents the mean number of cars.
The other options do not accurately describe the interpretation of the standard deviation:
- (a) is incorrect because the mean number of cars is not necessarily 1.08.
- (b) is a vague interpretation and doesn’t correctly describe the concept of standard deviation.
- (c) is misleading since the standard deviation doesn’t guarantee that all values are within 1.08 of the mean.
- (e) is incorrect because the mean number of cars is not described as being “about 1.08 from the expected value,” and the expected value is the mean.
Thus, option (d) correctly captures the meaning of the standard deviation in terms of the range within which most data points (households) will fall.