Mathematics, 24.01.2020 22:31, MarlonJ02

The lives of certain extra-life light bulbs are normally distributed with a mean equal to 1350 hours and a standard deviation equal to 18 hours1. what percentage of bulbs will have a life between 1350 and 1377 hr? 2. what percentage of bulbs will have a life between 1341 and 1350 hr? 3. what percentage of bulbs will have a life between 1338 and 1365 hr? 4. what percentage of bulbs will have a life between 1365 and 1377 hr?

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answered: a6igai1

1. 43.32% of bulbs will have a life between 1350 and 1377 hr.

2. 19.15% of bulbs will have a life between 1341 and 1350 hr.

3. 54.53% of bulbs will have a life between 1338 and 1365 hr.

4. 13.65% of bulbs will have a life between 1365 and 1377 hr.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 1350, \sigma = 18$

1. What percentage of bulbs will have a life between 1350 and 1377 hr?

This is the pvalue of Z when X = 1377 subtracted by the pvalue of Z when X = 1350.

X = 1377

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1377 - 1350}{18}$

Z = 1.5

Z = 1.5 has a pvalue of 0.9332.

X = 1350

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1350 - 1350}{18}$

Z = 0

Z = 0 has a pvalue of 0.5.

So 0.9332 - 0.5 = 0.4332 = 43.32% of bulbs will have a life between 1350 and 1377 hr.

2. What percentage of bulbs will have a life between 1341 and 1350 hr?

This is the pvalue of Z when X = 1350 subtracted by the pvalue of Z when X = 1341.

X = 1350

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1350 - 1350}{18}$

Z = 0

Z = 0 has a pvalue of 0.5.

X = 1341

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1341 - 1350}{18}$

Z = -0.5

Z = -0.5 has a pvalue of 0.3085.

So 0.5 - 0.3085 = 0.1915 = 19.15% of bulbs will have a life between 1341 and 1350 hr.

3. What percentage of bulbs will have a life between 1338 and 1365 hr?

This is the pvalue of Z when X = 1365 subtracted by the pvalue of Z when X = 1338.

X = 1365

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1365 - 1350}{18}$

Z = 0.83

Z = 0.83 has a pvalue of 0.7967

X = 1338

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1338 - 1350}{18}$

Z = -0.67

Z = -0.67 has a pvalue of 0.2514.

So 0.7967 - 0.2514 = 0.5453 = 54.53% of bulbs will have a life between 1338 and 1365 hr.

4. What percentage of bulbs will have a life between 1365 and 1377 hr?

This is the pvalue of Z when X = 1377 subtracted by the pvalue of Z when X = 1365.

X = 1377

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1377 - 1350}{18}$

Z = 1.5

Z = 1.5 has a pvalue of 0.9332.

X = 1365

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1365 - 1350}{18}$

Z = 0.83

Z = 0.83 has a pvalue of 0.7967.

So 0.9332 - 0.7967 = 0.1365 = 13.65% of bulbs will have a life between 1365 and 1377 hr.

answered: Guest

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answered: Guest

The answer is 49/16 or 3 1/16

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The lives of certain extra-life light bulbs are normally distributed with a mean equal to 1350 hours...