91Ó°ÊÓ

In the context of a binomial distribution, the **Expected Value** is a way to predict the average outcome of an event over a large number of trials. For this exercise, we're dealing with television tubes that have a 10% chance of burning out before their guarantee expires. To calculate the expected number of tubes that need replacement out of 1000 sold, we use the formula for expected value in a binomial distribution:

• Formula: \( E(Y) = n \times p \)
• Where \( n \) is the total number of trials (or tubes in this case) and \( p \) is the probability of success (or failure, depending on perspective).

Plugging in our values, \( n = 1000 \) and \( p = 0.1 \), we find \( E(Y) = 1000 \times 0.1 = 100 \). This means, on average, 100 tubes are expected to fail. This calculation helps us anticipate occurrences and plan replacements accordingly.

The **Variance** is a measure of how much the values in a distribution differ from the expected value. It provides insight into the variability or spread within the data. In a binomial distribution, variance can be calculated using the formula:

• Formula: \( Var(Y) = n \times p \times (1-p) \)
• Where \( p \) is the probability of success and \( 1-p \) is the probability of failure.

For our exercise, we calculate the variance with \( n = 1000 \), \( p = 0.1 \), and the complement \( 1-p = 0.9 \). This gives: \[Var(Y) = 1000 \times 0.1 \times 0.9 = 90\]A variance of 90 means that the number of tubes failing will fluctuate around the mean, showing how varied or spread-out the failures can be from the expected average.

The **Standard Deviation** is a measure derived from the variance and provides a clearer sense of the data spread by interpreting it in the same units as the original data. It's the square root of the variance:

• Formula: \( \text{SD}(Y) = \sqrt{Var(Y)} \)

With a variance of 90 calculated previously, our standard deviation is:\[\text{SD}(Y) = \sqrt{90} \approx 9.49\]This result indicates that on average, the number of failing tubes will vary by about 9.49 from the expected value of 100. Standard deviation provides a tangible way to assess variability and predict outcomes within the dataset.

The **Empirical Rule** is a key statistical concept that helps understand the spread of data in a normal distribution. It's often referred to as the "68-95-99.7 rule." It indicates how data corresponds around the mean in terms of standard deviations:

• 68% of data falls within one standard deviation (\(\pm 1\text{ SD}\))
• 95% within two (\(\pm 2\text{ SD}\))
• 99.7% within three (\(\pm 3\text{ SD}\))

In this problem, we use the empirical rule to anticipate the range in which the number of failing tubes will fall. Since we calculated the expected value as 100 and the standard deviation as approximately 9.49, we find our range by:\[E(Y) - \text{SD}(Y) \leq Y \leq E(Y) + \text{SD}(Y)\]This simplifies to approximately \( 90.51 \leq Y \leq 109.49 \). Rounding to whole numbers, we expect between 91 to 109 tubes to fail in about 68% of the cases. This rule thus helps predict real-life occurrences based on probabilistic models.