To determine the number of light bulbs expected to last between 2030 hours and 2060 hours, we need to calculate the z-scores corresponding to these values and then use the z-score formula to find the proportion of light bulbs within this range.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = value
μ = mean
σ = standard deviation
For 2030 hours:
z1 = (2030 - 2000) / 25
For 2060 hours:
z2 = (2060 - 2000) / 25
Now, we can use the z-scores to find the proportions associated with each value using a standard normal distribution table or calculator. The table or calculator will provide the area/proportion under the normal curve between the mean and each z-score.
Let's calculate the z-scores and find the proportions:
z1 = (2030 - 2000) / 25 = 1.2
z2 = (2060 - 2000) / 25 = 2.4
Using a standard normal distribution table or calculator, we can find the proportions corresponding to these z-scores:
P(z < 1.2) ≈ 0.8849
P(z < 2.4) ≈ 0.9918
To find the proportion of light bulbs expected to last between 2030 hours and 2060 hours, we subtract the cumulative probabilities:
P(2030 < x < 2060) = P(z1 < z < z2) = P(z < z2) - P(z < z1)
P(2030 < x < 2060) ≈ 0.9918 - 0.8849
Finally, we multiply this proportion by the total number of light bulbs (665) to get the estimated number of light bulbs expected to last between 2030 hours and 2060 hours:
Number of light bulbs ≈ (0.9918 - 0.8849) * 665
Rounding to the nearest whole number, the expected number of light bulbs that would last between 2030 hours and 2060 hours is approximately 71.
♥️
