The table lists the number of wireless devices per household in a small town in the United States.

c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.

검증된 단계별 안내

Step 1: Calculate the total number of households by summing the frequencies in the table. This will be used to compute the probabilities for the probability distribution. Total households = 277 + 471 + 243 + 105 + 46 + 22.

Step 2: Compute the probability for each number of wireless devices by dividing the frequency of each category by the total number of households. For example, P(0 devices) = 277 / Total households, P(1 device) = 471 / Total households, and so on.

Step 3: Calculate the mean (expected value) of the probability distribution using the formula: \( \mu = \sum (x \cdot P(x)) \), where \( x \) is the number of wireless devices and \( P(x) \) is the probability of \( x \). Multiply each \( x \) value by its corresponding probability and sum the results.

Step 4: Compute the variance using the formula: \( \sigma^2 = \sum ((x - \mu)^2 \cdot P(x)) \). Subtract the mean from each \( x \), square the result, multiply by the corresponding probability, and sum these values.

Step 5: Find the standard deviation by taking the square root of the variance: \( \sigma = \sqrt{\sigma^2} \). Interpret the mean, variance, and standard deviation in the context of the number of wireless devices per household.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mean

The mean, or average, is a measure of central tendency that summarizes a set of values by dividing the sum of all values by the number of values. In the context of a probability distribution, the mean represents the expected value, indicating the average number of wireless devices per household. It is calculated by multiplying each value by its probability and summing the results.

Variance

Variance is a statistical measure that quantifies the degree of spread in a set of values. It is calculated by taking the average of the squared differences from the mean. In a probability distribution, variance helps to understand how much the number of wireless devices per household varies from the mean, providing insight into the distribution's dispersion.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of the average distance of each data point from the mean. It is a key concept in statistics as it indicates the extent of variability in a dataset. A low standard deviation suggests that the data points are close to the mean, while a high standard deviation indicates a wider spread of values, which is crucial for interpreting the distribution of wireless devices in households.

The table lists the number of wireless devices per household in a small town in the United States.c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.