Math / Data & Statistics

QuestionA company installs 5,000 light bulbs. The lifetimes of the lightbulbs are approximately normally distributed with a mean of 500 hours and a standard deviation of 100 hours. Find the approximate number of bulbs that can be expected to last between 290 hours and 500 hours. A. 2,413 B. 2,911 C. 2,913 D. 2,410

Studdy Solution

STEP 1

1. The lifetimes of the light bulbs are normally distributed.
2. The mean lifetime is 500 hours.
3. The standard deviation of the lifetimes is 100 hours.
4. We need to find the number of bulbs lasting between 290 hours and 500 hours out of 5,000 bulbs.

STEP 2

1. Convert the given lifetimes to z-scores.
2. Use the standard normal distribution to find the probabilities.
3. Calculate the number of bulbs corresponding to the probability.

STEP 3

Convert the lower bound (290 hours) to a z-score using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 290$ , $\mu = 500$ , and $\sigma = 100$ .
$z = \frac{290 - 500}{100} = \frac{-210}{100} = -2.1$

STEP 4

Convert the upper bound (500 hours) to a z-score:
$z = \frac{500 - 500}{100} = 0$

STEP 5

Use the standard normal distribution table to find the probability corresponding to $z = -2.1$ .
The probability $P(Z < -2.1) \approx 0.0179$ .

STEP 6

Find the probability corresponding to $z = 0$ .
The probability $P(Z < 0) = 0.5$ .

STEP 7

Calculate the probability of a bulb lasting between 290 and 500 hours:
$P(-2.1 < Z < 0) = P(Z < 0) - P(Z < -2.1) = 0.5 - 0.0179 = 0.4821$

STEP 8

Calculate the number of bulbs expected to last between 290 and 500 hours:
$0.4821 \times 5000 \approx 2410.5$
Round to the nearest whole number: 2,410 bulbs.
The approximate number of bulbs expected to last between 290 hours and 500 hours is:
$\boxed{2,410}$

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Studdy Solution

STEP 1

1. The lifetimes of the light bulbs are normally distributed.
2. The mean lifetime is 500 hours.
3. The standard deviation of the lifetimes is 100 hours.
4. We need to find the number of bulbs lasting between 290 hours and 500 hours out of 5,000 bulbs.

STEP 2

1. Convert the given lifetimes to z-scores.
2. Use the standard normal distribution to find the probabilities.
3. Calculate the number of bulbs corresponding to the probability.

STEP 3

Convert the lower bound (290 hours) to a z-score using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 290$ , $\mu = 500$ , and $\sigma = 100$ .
$z = \frac{290 - 500}{100} = \frac{-210}{100} = -2.1$

STEP 4

Convert the upper bound (500 hours) to a z-score:
$z = \frac{500 - 500}{100} = 0$

STEP 5

Use the standard normal distribution table to find the probability corresponding to $z = -2.1$ .
The probability $P(Z < -2.1) \approx 0.0179$ .

STEP 6

Find the probability corresponding to $z = 0$ .
The probability $P(Z < 0) = 0.5$ .

STEP 7

Calculate the probability of a bulb lasting between 290 and 500 hours:
$P(-2.1 < Z < 0) = P(Z < 0) - P(Z < -2.1) = 0.5 - 0.0179 = 0.4821$

STEP 8

Calculate the number of bulbs expected to last between 290 and 500 hours:
$0.4821 \times 5000 \approx 2410.5$
Round to the nearest whole number: 2,410 bulbs.
The approximate number of bulbs expected to last between 290 hours and 500 hours is:
$\boxed{2,410}$

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Studdy Solution

STEP 1

1. The lifetimes of the light bulbs are normally distributed.2. The mean lifetime is 500 hours.3. The standard deviation of the lifetimes is 100 hours.4. We need to find the number of bulbs lasting between 290 hours and 500 hours out of 5,000 bulbs.

STEP 2

1. Convert the given lifetimes to z-scores.2. Use the standard normal distribution to find the probabilities.3. Calculate the number of bulbs corresponding to the probability.

STEP 3

STEP 4

Convert the upper bound (500 hours) to a z-score:z=500−500100=0 z = \frac{500 - 500}{100} = 0 z=100500−500​=0

STEP 5

Use the standard normal distribution table to find the probability corresponding to z=−2.1 z = -2.1 z=−2.1.The probability P(Z<−2.1)≈0.0179 P(Z < -2.1) \approx 0.0179 P(Z<−2.1)≈0.0179.

STEP 6

Find the probability corresponding to z=0 z = 0 z=0.The probability P(Z<0)=0.5 P(Z < 0) = 0.5 P(Z<0)=0.5.

STEP 7

Calculate the probability of a bulb lasting between 290 and 500 hours:P(−2.1<Z<0)=P(Z<0)−P(Z<−2.1)=0.5−0.0179=0.4821 P(-2.1 < Z < 0) = P(Z < 0) - P(Z < -2.1) = 0.5 - 0.0179 = 0.4821 P(−2.1<Z<0)=P(Z<0)−P(Z<−2.1)=0.5−0.0179=0.4821

STEP 8

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1. The lifetimes of the light bulbs are normally distributed.
2. The mean lifetime is 500 hours.
3. The standard deviation of the lifetimes is 100 hours.
4. We need to find the number of bulbs lasting between 290 hours and 500 hours out of 5,000 bulbs.

1. Convert the given lifetimes to z-scores.
2. Use the standard normal distribution to find the probabilities.
3. Calculate the number of bulbs corresponding to the probability.

Convert the upper bound (500 hours) to a z-score:
$z = \frac{500 - 500}{100} = 0$

Use the standard normal distribution table to find the probability corresponding to $z = -2.1$ .
The probability $P(Z < -2.1) \approx 0.0179$ .

Find the probability corresponding to $z = 0$ .
The probability $P(Z < 0) = 0.5$ .

Calculate the probability of a bulb lasting between 290 and 500 hours:
$P(-2.1 < Z < 0) = P(Z < 0) - P(Z < -2.1) = 0.5 - 0.0179 = 0.4821$