Math / Data & Statistics

QuestionThe lifespan of some macaws in captivity is normally distributed. Their lifespan has a mean of 50 years and a standard deviation of 8 years.
What is the probability of a macaw with a lifespan greater than 60 years?

Studdy Solution

STEP 1

1. The lifespan of macaws is normally distributed with a mean ( $\mu$ ) of 50 years.
2. The standard deviation ( $\sigma$ ) of the lifespan is 8 years.
3. We are looking for the probability that a macaw's lifespan is greater than 60 years.

STEP 2

1. Calculate the z-score for a lifespan of 60 years.
2. Use the z-score to find the probability of a lifespan greater than 60 years.

STEP 3

Calculate the z-score using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 60$ , $\mu = 50$ , and $\sigma = 8$ .
$z = \frac{60 - 50}{8} = \frac{10}{8} = 1.25$

STEP 4

Use the z-score to find the probability. Look up the z-score of 1.25 in the standard normal distribution table to find the probability that a macaw's lifespan is less than 60 years.
The probability that $Z < 1.25$ is approximately 0.8944.

STEP 5

Since we want the probability of a lifespan greater than 60 years, we calculate:
$P(X > 60) = 1 - P(Z < 1.25)$
$P(X > 60) = 1 - 0.8944 = 0.1056$
The probability of a macaw with a lifespan greater than 60 years is approximately:
$\boxed{0.1056}$

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QuestionThe lifespan of some macaws in captivity is normally distributed. Their lifespan has a mean of 50 years and a standard deviation of 8 years.
What is the probability of a macaw with a lifespan greater than 60 years?

Studdy Solution

STEP 1

1. The lifespan of macaws is normally distributed with a mean ( $\mu$ ) of 50 years.
2. The standard deviation ( $\sigma$ ) of the lifespan is 8 years.
3. We are looking for the probability that a macaw's lifespan is greater than 60 years.

STEP 2

1. Calculate the z-score for a lifespan of 60 years.
2. Use the z-score to find the probability of a lifespan greater than 60 years.

STEP 3

Calculate the z-score using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 60$ , $\mu = 50$ , and $\sigma = 8$ .
$z = \frac{60 - 50}{8} = \frac{10}{8} = 1.25$

STEP 4

Use the z-score to find the probability. Look up the z-score of 1.25 in the standard normal distribution table to find the probability that a macaw's lifespan is less than 60 years.
The probability that $Z < 1.25$ is approximately 0.8944.

STEP 5

Since we want the probability of a lifespan greater than 60 years, we calculate:
$P(X > 60) = 1 - P(Z < 1.25)$
$P(X > 60) = 1 - 0.8944 = 0.1056$
The probability of a macaw with a lifespan greater than 60 years is approximately:
$\boxed{0.1056}$

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QuestionThe lifespan of some macaws in captivity is normally distributed. Their lifespan has a mean of 50 years and a standard deviation of 8 years.What is the probability of a macaw with a lifespan greater than 60 years?

Studdy Solution

STEP 1

1. The lifespan of macaws is normally distributed with a mean (μ\muμ) of 50 years.2. The standard deviation (σ\sigmaσ) of the lifespan is 8 years.3. We are looking for the probability that a macaw's lifespan is greater than 60 years.

STEP 2

1. Calculate the z-score for a lifespan of 60 years.2. Use the z-score to find the probability of a lifespan greater than 60 years.

STEP 3

Calculate the z-score using the formula:z=X−μσ z = \frac{X - \mu}{\sigma} z=σX−μ​where X=60 X = 60 X=60, μ=50 \mu = 50 μ=50, and σ=8 \sigma = 8 σ=8.z=60−508=108=1.25 z = \frac{60 - 50}{8} = \frac{10}{8} = 1.25 z=860−50​=810​=1.25

STEP 4

Use the z-score to find the probability. Look up the z-score of 1.25 in the standard normal distribution table to find the probability that a macaw's lifespan is less than 60 years.The probability that Z<1.25 Z < 1.25 Z<1.25 is approximately 0.8944.

STEP 5

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe lifespan of some macaws in captivity is normally distributed. Their lifespan has a mean of 50 years and a standard deviation of 8 years.
What is the probability of a macaw with a lifespan greater than 60 years?

1. The lifespan of macaws is normally distributed with a mean ( $\mu$ ) of 50 years.
2. The standard deviation ( $\sigma$ ) of the lifespan is 8 years.
3. We are looking for the probability that a macaw's lifespan is greater than 60 years.

1. Calculate the z-score for a lifespan of 60 years.
2. Use the z-score to find the probability of a lifespan greater than 60 years.

Calculate the z-score using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 60$ , $\mu = 50$ , and $\sigma = 8$ .
$z = \frac{60 - 50}{8} = \frac{10}{8} = 1.25$

Use the z-score to find the probability. Look up the z-score of 1.25 in the standard normal distribution table to find the probability that a macaw's lifespan is less than 60 years.
The probability that $Z < 1.25$ is approximately 0.8944.