Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. Seventy percent of all light bulbs last at least how long?

Using the exponential distribution, it is found that 70% of all light bulbs last at least 2.85 years.

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:  

In which is the decay parameter.

The probability that x is lower or equal to a is given by:

Which has the following solution:

The probability of finding a value higher than x is:

In this problem, the mean lifetime is of 8 years, hence .

Seventy percent of all light bulbs last at least x, considering that .

Hence:

70% of all light bulbs last at least 2.85 years.

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Part A: There is a correlation between the number of workers in a factory and the number of units produced daily.

Part B: The function that best fits the data is

Part C: The slope indicates the rate of change of number of units produced daily(y) over change in number of workers(x). The y-intercept of the plot indicates the number of units produced by number of workers.

Explanation:

Part A:  Let us determine it using the slope formula which is given by,

Substituting, the coordinates and .

Thus, we have,

Thus, slope = 5

The y-intercept can be determined when the value of x is zero.

Hence, y-intercept b = 2

Substituting these values in the slope-intercept formula,

Thus, there is a correlation between the number of workers in a factory and the number of units produced daily.

Part B: The function that best fits the data is given by

The explanation for the function equation is provided above.

Part C:

The slope is

The slope indicates the rate of change of number of units produced daily(y) over change in number of workers(x).

The y-intercept is

The y-intercept indicates the number of units produced by number of workers.