Math / Data & Statistics

QuestionPeriod
7. In a normal distribution, $x=3$ and $z=0.67$ . This tells you that $x=3$ is $\qquad$ standard deviations to the $\qquad$ (left or right) of the mean.
8. In a normal distribution, $x=-5$ and $z=-3.14$ . This tells you that $x=-5$ is $\qquad$ mean. standard deviations to the $\qquad$ (left or right) of the
9. The life of Sunshine DVD players is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A DVD player is guaranteed for three years. We are interested in the length of time a DVD player lasts. Find the z score corresponding to the guaranteed life of 3 years
10. The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean 125 and standard deviation 14. Systolic blood pressure for males follows a normal distribution. a. Calculate the z-scores for the male systolic blood pressure 100 and 150 millimeters. b. If a male friend of yours said that he thought his systolic blood pressure was 2.5 standard deviations below the mean, but he believed his blood pressure was between 100 and 150 millimeters, what would you say to him.

Studdy Solution

STEP 1

1. The z-score formula is $z = \frac{x - \mu}{\sigma}$ , where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.
2. Positive z-scores indicate values to the right of the mean, while negative z-scores indicate values to the left of the mean.
3. The mean and standard deviation for the DVD player problem are given as 4.1 years and 1.3 years, respectively.
4. The mean and standard deviation for the systolic blood pressure problem are given as 125 mm and 14 mm, respectively.

STEP 2

1. Determine the direction and number of standard deviations for given z-scores.
2. Calculate the z-score for the guaranteed life of a DVD player.
3. Calculate z-scores for given systolic blood pressures.
4. Analyze the friend's claim about his systolic blood pressure.

STEP 3

For $x = 3$ and $z = 0.67$ , since $z$ is positive, $x = 3$ is 0.67 standard deviations to the right of the mean.

STEP 4

For $x = -5$ and $z = -3.14$ , since $z$ is negative, $x = -5$ is 3.14 standard deviations to the left of the mean.

STEP 5

Calculate the z-score for the guaranteed life of 3 years using the formula:
$z = \frac{3 - 4.1}{1.3} = \frac{-1.1}{1.3} \approx -0.846$

STEP 6

Calculate the z-score for a systolic blood pressure of 100 mm:
$z = \frac{100 - 125}{14} = \frac{-25}{14} \approx -1.79$
Calculate the z-score for a systolic blood pressure of 150 mm:
$z = \frac{150 - 125}{14} = \frac{25}{14} \approx 1.79$

STEP 7

If the friend believes his systolic blood pressure is 2.5 standard deviations below the mean, calculate the blood pressure:
$x = 125 + (-2.5 \times 14) = 125 - 35 = 90$
Since 90 mm is not between 100 mm and 150 mm, inform your friend that his belief is inconsistent with his stated range.
The solutions are:
1. $x = 3$ is 0.67 standard deviations to the right of the mean.
2. $x = -5$ is 3.14 standard deviations to the left of the mean.
3. The z-score for 3 years is approximately $-0.846$ .
4. The z-scores for 100 mm and 150 mm are approximately $-1.79$ and $1.79$ , respectively.
5. The friend's belief about his blood pressure is inconsistent with his stated range.

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Studdy Solution

STEP 1

STEP 2

1. Determine the direction and number of standard deviations for given z-scores.
2. Calculate the z-score for the guaranteed life of a DVD player.
3. Calculate z-scores for given systolic blood pressures.
4. Analyze the friend's claim about his systolic blood pressure.

STEP 3

For $x = 3$ and $z = 0.67$ , since $z$ is positive, $x = 3$ is 0.67 standard deviations to the right of the mean.

STEP 4

For $x = -5$ and $z = -3.14$ , since $z$ is negative, $x = -5$ is 3.14 standard deviations to the left of the mean.

STEP 5

Calculate the z-score for the guaranteed life of 3 years using the formula:
$z = \frac{3 - 4.1}{1.3} = \frac{-1.1}{1.3} \approx -0.846$

STEP 6

Calculate the z-score for a systolic blood pressure of 100 mm:
$z = \frac{100 - 125}{14} = \frac{-25}{14} \approx -1.79$
Calculate the z-score for a systolic blood pressure of 150 mm:
$z = \frac{150 - 125}{14} = \frac{25}{14} \approx 1.79$

STEP 7

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Studdy Solution

STEP 1

STEP 2

1. Determine the direction and number of standard deviations for given z-scores.2. Calculate the z-score for the guaranteed life of a DVD player.3. Calculate z-scores for given systolic blood pressures.4. Analyze the friend's claim about his systolic blood pressure.

STEP 3

For x=3 x = 3 x=3 and z=0.67 z = 0.67 z=0.67, since z z z is positive, x=3 x = 3 x=3 is 0.67 standard deviations to the right of the mean.

STEP 4

For x=−5 x = -5 x=−5 and z=−3.14 z = -3.14 z=−3.14, since z z z is negative, x=−5 x = -5 x=−5 is 3.14 standard deviations to the left of the mean.

STEP 5

Calculate the z-score for the guaranteed life of 3 years using the formula:z=3−4.11.3=−1.11.3≈−0.846 z = \frac{3 - 4.1}{1.3} = \frac{-1.1}{1.3} \approx -0.846 z=1.33−4.1​=1.3−1.1​≈−0.846

STEP 6

STEP 7

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1. Determine the direction and number of standard deviations for given z-scores.
2. Calculate the z-score for the guaranteed life of a DVD player.
3. Calculate z-scores for given systolic blood pressures.
4. Analyze the friend's claim about his systolic blood pressure.

For $x = 3$ and $z = 0.67$ , since $z$ is positive, $x = 3$ is 0.67 standard deviations to the right of the mean.

For $x = -5$ and $z = -3.14$ , since $z$ is negative, $x = -5$ is 3.14 standard deviations to the left of the mean.

Calculate the z-score for the guaranteed life of 3 years using the formula:
$z = \frac{3 - 4.1}{1.3} = \frac{-1.1}{1.3} \approx -0.846$