I need help asap!!!!!!!!!!!!!!!!!!!!

Andrew plans to retire in 32 years. He plans to invest part of his retirement funds in stocks, so he seeks out information on pa

Debora [2.8K]

Answer:

a) 0.0885 = 8.85% probability that the mean annual return on common stocks over the next 40 years will exceed 13%.

b) 0.4129 = 41.29% probability that the mean return will be less than 8%

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean 8.7% and standard deviation 20.2%.

This means that $\mu = 8.7, \sigma = 20.2$

40 years:

This means that $n = 40, s = \frac{20.2}{\sqrt{40}}$

(a) What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40 years will exceed 13%?

This is 1 subtracted by the pvalue of Z when X = 13. So

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

By the Central Limit Theorem

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

$Z = \frac{13 - 8.7}{\frac{20.2}{\sqrt{40}}}$

$Z = 1.35$

$Z = 1.35$ has a pvalue of 0.9115

1 - 0.9115 = 0.0885

0.0885 = 8.85% probability that the mean annual return on common stocks over the next 40 years will exceed 13%.

(b) What is the probability that the mean return will be less than 8%?

This is the pvalue of Z when X = 8. So

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

$Z = \frac{8 - 8.7}{\frac{20.2}{\sqrt{40}}}$

$Z = -0.22$

$Z = -0.22$ has a pvalue of 0.4129

0.4129 = 41.29% probability that the mean return will be less than 8%

8 0

3 years ago

The caller times at a customer service center has an exponential distribution with an average of 10 seconds. Find the probabilit

ololo11 [35]

Answer:

0.9179 = 91.79% probability that a randomly selected call time will be less than 25 seconds

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

$f(x) = \mu e^{-\mu x}$

In which $\mu = \frac{1}{m}$ is the decay parameter.

The probability that x is lower or equal to a is given by:

$P(X \leq x) = \int\limits^a_0 {f(x)} \, dx$

Which has the following solution:

$P(X \leq x) = 1 - e^{-\mu x}$

The probability of finding a value higher than x is:

$P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}$

In this question:

$m = 10, \mu = \frac{1}{10} = 0.1$

Find the probability that a randomly selected call time will be less than 25 seconds?

$P(X \leq x) = 1 - e^{-\mu x}$

$P(X \leq 25) = 1 - e^{-0.1*25} = 0.9179$

0.9179 = 91.79% probability that a randomly selected call time will be less than 25 seconds

4 0

3 years ago

compare the two numbers to find which is greater. explain how you can compare them without writing them in standard notation fir

Anna71 [15]

The number with the largest exponent is the largest value overall. In this case, that would be 2.1 x 10^8 which is the largest value since 8 is the largest exponent.

Note: The number 4.5 x 10^6 is the number 4.5 million, while the number 2.1 x 10^8 is the number 210 million. The positive exponent means "move the decimal point that number of times to the right". So 4.5 x 10^6 means "move the decimal point 6 spots to the right" so you can convert to standard form. The larger the exponent, the more places the decimal point moves to the right.

5 0

4 years ago

How can i solve for x: 64=16 x^2

Mariana [72]

$64=16x^2\\
x^2=4\\
x=-2\vee x=2$