Math / Data & Statistics

QuestionRisk taking is an important part of investing. In order to make suitable investment decisions on behalf of their customers, portfolio managers give a questionnaire to new customers to measure their desire to take financial risks. The scores on the questionnaire are approximately normally distributed with a mean of 51 and a standard deviation of 16 . The customers with scores in the bottom $15 \%$ are described as "risk averse." What is the questionnaire score that separates customers who are considered risk averse from those who are not? Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place. $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the score that separates the bottom 15% of customers (the risk-averse ones) from the rest. Watch out! Don't forget that we're dealing with a normal distribution, so we'll need to use the z-score and look up values in a z-table!

STEP 2

1. Find the z-score.
2. Calculate the risk averse score.

STEP 3

We're looking for the z-score that corresponds to the 15th percentile of the standard normal distribution.
This z-score represents the point below which 15% of the data falls.

STEP 4

To find this z-score, we can use a z-table (or a calculator).
We're looking for the closest probability to 0.15 in the body of the z-table.
This value helps us link the percentage to a specific z-score.

STEP 5

Looking it up, we find that the closest probability is 0.1492, which corresponds to a z-score of approximately -1.04.
This means that a score of -1.04 standard deviations below the mean separates the bottom 15% from the rest.

STEP 6

Now that we have our z-score (-1.04), we can use the z-score formula to find the actual questionnaire score that corresponds to this percentile.
Remember, the z-score formula is:
$z = \frac{x - \mu}{\sigma}$ where $z$ is the z-score, $x$ is the raw score we want to find, $\mu$ is the mean, and $\sigma$ is the standard deviation.

STEP 7

We know the following values: * $z = \mathbf{-1.04}$ (from the z-table) * $\mu = \mathbf{51}$ (the mean score) * $\sigma = \mathbf{16}$ (the standard deviation)

STEP 8

Plug these values into the z-score formula:
$-1.04 = \frac{x - 51}{16}$

STEP 9

To solve for $x$ , we first multiply both sides of the equation by 16:
$-1.04 \cdot 16 = x - 51$ $-16.64 = x - 51$

STEP 10

Then, we add 51 to both sides of the equation:
$-16.64 + 51 = x$ $x = 34.36$

STEP 11

Finally, we round our answer to one decimal place, as requested in the problem:
$x \approx 34.4$

STEP 12

The questionnaire score that separates risk-averse customers from the others is approximately 34.4.

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Studdy Solution

STEP 1

What is this asking? We need to find the score that separates the bottom 15% of customers (the risk-averse ones) from the rest. Watch out! Don't forget that we're dealing with a normal distribution, so we'll need to use the z-score and look up values in a z-table!

STEP 2

1. Find the z-score.
2. Calculate the risk averse score.

STEP 3

We're looking for the z-score that corresponds to the 15th percentile of the standard normal distribution.
This z-score represents the point below which 15% of the data falls.

STEP 4

To find this z-score, we can use a z-table (or a calculator).
We're looking for the closest probability to 0.15 in the body of the z-table.
This value helps us link the percentage to a specific z-score.

STEP 5

Looking it up, we find that the closest probability is 0.1492, which corresponds to a z-score of approximately -1.04.
This means that a score of -1.04 standard deviations below the mean separates the bottom 15% from the rest.

STEP 6

STEP 7

We know the following values: * $z = \mathbf{-1.04}$ (from the z-table) * $\mu = \mathbf{51}$ (the mean score) * $\sigma = \mathbf{16}$ (the standard deviation)

STEP 8

Plug these values into the z-score formula:
$-1.04 = \frac{x - 51}{16}$

STEP 9

To solve for $x$ , we first multiply both sides of the equation by 16:
$-1.04 \cdot 16 = x - 51$ $-16.64 = x - 51$

STEP 10

Then, we add 51 to both sides of the equation:
$-16.64 + 51 = x$ $x = 34.36$

STEP 11

Finally, we round our answer to one decimal place, as requested in the problem:
$x \approx 34.4$

STEP 12

The questionnaire score that separates risk-averse customers from the others is approximately 34.4.

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Studdy Solution

STEP 1

What is this asking? We need to find the score that separates the bottom 15% of customers (the risk-averse ones) from the rest. Watch out! Don't forget that we're dealing with a normal distribution, so we'll need to use the z-score and look up values in a z-table!

STEP 2

1. Find the z-score.2. Calculate the risk averse score.

STEP 3

We're looking for the z-score that corresponds to the **15th percentile** of the standard normal distribution.This z-score represents the point below which 15% of the data falls.

STEP 4

To find this z-score, we can use a z-table (or a calculator).We're looking for the closest probability to **0.15** in the body of the z-table.This value helps us link the percentage to a specific z-score.

STEP 5

Looking it up, we find that the closest probability is **0.1492**, which corresponds to a z-score of approximately **-1.04**.This means that a score of -1.04 standard deviations below the mean separates the bottom 15% from the rest.

STEP 6

STEP 7

We know the following values: * z=−1.04z = \mathbf{-1.04}z=−1.04 (from the z-table) * μ=51\mu = \mathbf{51}μ=51 (the mean score) * σ=16\sigma = \mathbf{16}σ=16 (the standard deviation)

STEP 8

**Plug these values** into the z-score formula:−1.04=x−5116 -1.04 = \frac{x - 51}{16} −1.04=16x−51​

STEP 9

To **solve for** xxx, we first **multiply** both sides of the equation by 16:−1.04⋅16=x−51 -1.04 \cdot 16 = x - 51 −1.04⋅16=x−51−16.64=x−51 -16.64 = x - 51 −16.64=x−51

STEP 10

Then, we **add 51** to both sides of the equation:−16.64+51=x -16.64 + 51 = x −16.64+51=xx=34.36 x = 34.36 x=34.36

STEP 11

Finally, we **round** our answer to one decimal place, as requested in the problem:x≈34.4 x \approx 34.4 x≈34.4

STEP 12

The questionnaire score that separates risk-averse customers from the others is approximately **34.4**.

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Studdy solves anything!

Start learning now

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

1. Find the z-score.
2. Calculate the risk averse score.

We're looking for the z-score that corresponds to the 15th percentile of the standard normal distribution.
This z-score represents the point below which 15% of the data falls.

To find this z-score, we can use a z-table (or a calculator).
We're looking for the closest probability to 0.15 in the body of the z-table.
This value helps us link the percentage to a specific z-score.

Looking it up, we find that the closest probability is 0.1492, which corresponds to a z-score of approximately -1.04.
This means that a score of -1.04 standard deviations below the mean separates the bottom 15% from the rest.

We know the following values: * $z = \mathbf{-1.04}$ (from the z-table) * $\mu = \mathbf{51}$ (the mean score) * $\sigma = \mathbf{16}$ (the standard deviation)

Plug these values into the z-score formula:
$-1.04 = \frac{x - 51}{16}$

To solve for $x$ , we first multiply both sides of the equation by 16:
$-1.04 \cdot 16 = x - 51$ $-16.64 = x - 51$

Then, we add 51 to both sides of the equation:
$-16.64 + 51 = x$ $x = 34.36$

Finally, we round our answer to one decimal place, as requested in the problem:
$x \approx 34.4$

The questionnaire score that separates risk-averse customers from the others is approximately 34.4.