Math / Data & Statistics

QuestionEvaluate the following probabilities based on the standard normal distribution:
Round all answers to at least 3 decimal places. a. $P(z<2.36)=$ $\square$ b. $P(z>-1.23)=$ $\square$ c. $P(-0.74<z<2.19)=$ $\square$ d. $P(z>0.5)=$ $\square$

Studdy Solution

STEP 1

1. The variable $z$ follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. Probabilities will be found using the standard normal distribution table or a calculator with statistical functions.

STEP 2

1. Calculate $P(z < 2.36)$ .
2. Calculate $P(z > -1.23)$ .
3. Calculate $P(-0.74 < z < 2.19)$ .
4. Calculate $P(z > 0.5)$ .

STEP 3

Use the standard normal distribution table or calculator to find $P(z < 2.36)$ .
$P(z < 2.36) \approx 0.9909$

STEP 4

To find $P(z > -1.23)$ , use the identity $P(z > a) = 1 - P(z < a)$ .
First, find $P(z < -1.23)$ .
$P(z < -1.23) \approx 0.1093$
Then calculate:
$P(z > -1.23) = 1 - P(z < -1.23) \approx 1 - 0.1093 = 0.8907$

STEP 5

To find $P(-0.74 < z < 2.19)$ , use the formula:
$P(a < z < b) = P(z < b) - P(z < a)$
First, find $P(z < 2.19)$ .
$P(z < 2.19) \approx 0.9857$
Then find $P(z < -0.74)$ .
$P(z < -0.74) \approx 0.2296$
Calculate:
$P(-0.74 < z < 2.19) = P(z < 2.19) - P(z < -0.74) \approx 0.9857 - 0.2296 = 0.7561$

STEP 6

To find $P(z > 0.5)$ , use the identity $P(z > a) = 1 - P(z < a)$ .
First, find $P(z < 0.5)$ .
$P(z < 0.5) \approx 0.6915$
Then calculate:
$P(z > 0.5) = 1 - P(z < 0.5) \approx 1 - 0.6915 = 0.3085$
The probabilities are: a. $P(z < 2.36) \approx 0.9909$ b. $P(z > -1.23) \approx 0.8907$ c. $P(-0.74 < z < 2.19) \approx 0.7561$ d. $P(z > 0.5) \approx 0.3085$

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QuestionEvaluate the following probabilities based on the standard normal distribution:
Round all answers to at least 3 decimal places. a. $P(z<2.36)=$ $\square$ b. $P(z>-1.23)=$ $\square$ c. $P(-0.74<z<2.19)=$ $\square$ d. $P(z>0.5)=$ $\square$

Studdy Solution

STEP 1

1. The variable $z$ follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. Probabilities will be found using the standard normal distribution table or a calculator with statistical functions.

STEP 2

1. Calculate $P(z < 2.36)$ .
2. Calculate $P(z > -1.23)$ .
3. Calculate $P(-0.74 < z < 2.19)$ .
4. Calculate $P(z > 0.5)$ .

STEP 3

Use the standard normal distribution table or calculator to find $P(z < 2.36)$ .
$P(z < 2.36) \approx 0.9909$

STEP 4

To find $P(z > -1.23)$ , use the identity $P(z > a) = 1 - P(z < a)$ .
First, find $P(z < -1.23)$ .
$P(z < -1.23) \approx 0.1093$
Then calculate:
$P(z > -1.23) = 1 - P(z < -1.23) \approx 1 - 0.1093 = 0.8907$

STEP 5

To find $P(-0.74 < z < 2.19)$ , use the formula:
$P(a < z < b) = P(z < b) - P(z < a)$
First, find $P(z < 2.19)$ .
$P(z < 2.19) \approx 0.9857$
Then find $P(z < -0.74)$ .
$P(z < -0.74) \approx 0.2296$
Calculate:
$P(-0.74 < z < 2.19) = P(z < 2.19) - P(z < -0.74) \approx 0.9857 - 0.2296 = 0.7561$

STEP 6

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Studdy Solution

STEP 1

1. The variable z z z follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1.2. Probabilities will be found using the standard normal distribution table or a calculator with statistical functions.

STEP 2

1. Calculate P(z<2.36) P(z < 2.36) P(z<2.36).2. Calculate P(z>−1.23) P(z > -1.23) P(z>−1.23).3. Calculate P(−0.74<z<2.19) P(-0.74 < z < 2.19) P(−0.74<z<2.19).4. Calculate P(z>0.5) P(z > 0.5) P(z>0.5).

STEP 3

Use the standard normal distribution table or calculator to find P(z<2.36) P(z < 2.36) P(z<2.36).P(z<2.36)≈0.9909 P(z < 2.36) \approx 0.9909 P(z<2.36)≈0.9909

STEP 4

STEP 5

STEP 6

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1. The variable $z$ follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
2. Probabilities will be found using the standard normal distribution table or a calculator with statistical functions.

1. Calculate $P(z < 2.36)$ .
2. Calculate $P(z > -1.23)$ .
3. Calculate $P(-0.74 < z < 2.19)$ .
4. Calculate $P(z > 0.5)$ .

Use the standard normal distribution table or calculator to find $P(z < 2.36)$ .
$P(z < 2.36) \approx 0.9909$