An electrical firm which manufactures a certain type of bulb wants to estimate its mean life. Assuming that the life of the light bulb is normally distributed and that the standard deviation is known to be 40 hours, how many bulbs should be tested so that we can be 90 percent confident that the estimate of the mean will not differ from the true mean life by more than 10 hours?
7
44
8
62
87
To determine how many bulbs should be tested, we can use the formula for sample size estimation for means:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level
σ = standard deviation
E = maximum allowable error
In this case, we want to be 90% confident that the estimate of the mean will not differ from the true mean life by more than 10 hours. So our desired confidence level is 90% or 0.9, and the maximum allowable error is 10 hours.
First, we need to find the Z-score for a 90% confidence level. We can use a Z-table or a calculator to find this value. The Z-score for a 90% confidence level is approximately 1.645.
Substituting the given values into the formula:
n = (1.645 * 40 / 10)^2
n = (65.8)^2
n ≈ 4339.64
Since we can't test a fraction of a bulb, we need to round up the sample size to the nearest whole number. Therefore, we should test at least 4340 bulbs.
So, the correct answer is 87.
To answer this question, we can use the formula for sample size calculation for estimating a population mean. The formula is given as:
n = (Z * σ / E)^2
Where:
- n = sample size
- Z = Z-score corresponding to the desired level of confidence (usually obtained from a Z-table)
- σ = standard deviation of the population
- E = maximum allowable error (also known as margin of error)
In this case, we want to be 90% confident, so the Z-score corresponding to a 90% confidence level is 1.645 (obtained from a Z-table).
The standard deviation (σ) is given as 40 hours.
The maximum allowable error (E) is 10 hours.
Plugging these values into the formula:
n = (1.645 * 40 / 10)^2
n = 6.58^2
n = 43.27
Since we can't have a fraction of a bulb, we need to round up to the nearest whole number. Therefore, the minimum number of bulbs that should be tested is 44.
So, the correct answer is 44.
Formula:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be found using a z-table to represent the 90% confidence interval, sd = 40, E = 10, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.