QuestionFor a distribution $N(5.43,0.54)$ , find the proportion of observations less than 5.05 and 5.79, rounded to four decimal places.

Studdy Solution

STEP 1

Assumptions1. The distribution is Normal with a mean of5.43 and a standard deviation of0.54. . We are looking for the proportion of observations less than5.05 and5.79.

STEP 2

First, we need to standardize the values5.05 and5.79 using the formula for the z-score. The z-score is a measure of how many standard deviations an element is from the mean.
$Z = \frac{X - \mu}{\sigma}$ where- X is the value from the dataset, - $\mu$ is the mean of the dataset, - $\sigma$ is the standard deviation of the dataset.

STEP 3

Now, plug in the given values for the first value (5.05), mean and standard deviation to calculate the z-score.
$Z_{5.05} = \frac{5.05 -5.43}{0.54}$

STEP 4

Calculate the z-score for the value.05.
$Z_{.05} = \frac{.05 -.43}{0.54} = -0.7037$

STEP 5

Now, plug in the given values for the second value (5.79), mean and standard deviation to calculate the z-score.
$Z_{5.79} = \frac{5.79 -5.43}{0.54}$

STEP 6

Calculate the z-score for the value5.79.
$Z_{5.79} = \frac{5.79 -5.43}{0.54} =0.666$

STEP 7

The z-scores represent the distance between the mean and the given values in terms of standard deviations. Now, we need to find the proportion of observations that are less than these z-scores. We can do this by looking up the z-scores in a standard normal distribution table or using a calculator with a normal distribution function.

STEP 8

Look up the z-score for5.05 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z < -0.7037)$

STEP 9

Calculate the proportion of observations less than5.05.
$(Z < -.7037) =.2408$

STEP 10

Look up the z-score for5.79 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z <0.6667)$

STEP 11

Calculate the proportion of observations less than5.79.
$(Z <0.6667) =0.7525$ The proportion of observations less than5.05 is0.2408 and the proportion of observations less than5.79 is0.7525.

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QuestionFor a distribution $N(5.43,0.54)$ , find the proportion of observations less than 5.05 and 5.79, rounded to four decimal places.

Studdy Solution

STEP 1

Assumptions1. The distribution is Normal with a mean of5.43 and a standard deviation of0.54. . We are looking for the proportion of observations less than5.05 and5.79.

STEP 2

STEP 3

Now, plug in the given values for the first value (5.05), mean and standard deviation to calculate the z-score.
$Z_{5.05} = \frac{5.05 -5.43}{0.54}$

STEP 4

Calculate the z-score for the value.05.
$Z_{.05} = \frac{.05 -.43}{0.54} = -0.7037$

STEP 5

Now, plug in the given values for the second value (5.79), mean and standard deviation to calculate the z-score.
$Z_{5.79} = \frac{5.79 -5.43}{0.54}$

STEP 6

Calculate the z-score for the value5.79.
$Z_{5.79} = \frac{5.79 -5.43}{0.54} =0.666$

STEP 7

STEP 8

Look up the z-score for5.05 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z < -0.7037)$

STEP 9

Calculate the proportion of observations less than5.05.
$(Z < -.7037) =.2408$

STEP 10

Look up the z-score for5.79 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z <0.6667)$

STEP 11

Calculate the proportion of observations less than5.79.
$(Z <0.6667) =0.7525$ The proportion of observations less than5.05 is0.2408 and the proportion of observations less than5.79 is0.7525.

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QuestionFor a distribution N(5.43,0.54)N(5.43,0.54)N(5.43,0.54), find the proportion of observations less than 5.05 and 5.79, rounded to four decimal places.

Studdy Solution

STEP 1

Assumptions1. The distribution is Normal with a mean of5.43 and a standard deviation of0.54. . We are looking for the proportion of observations less than5.05 and5.79.

STEP 2

STEP 3

Now, plug in the given values for the first value (5.05), mean and standard deviation to calculate the z-score.Z5.05=5.05−5.430.54Z_{5.05} = \frac{5.05 -5.43}{0.54}Z5.05​=0.545.05−5.43​

STEP 4

Calculate the z-score for the value.05.Z.05=.05−.430.54=−0.7037Z_{.05} = \frac{.05 -.43}{0.54} = -0.7037Z.05​=0.54.05−.43​=−0.7037

STEP 5

Now, plug in the given values for the second value (5.79), mean and standard deviation to calculate the z-score.Z5.79=5.79−5.430.54Z_{5.79} = \frac{5.79 -5.43}{0.54}Z5.79​=0.545.79−5.43​

STEP 6

Calculate the z-score for the value5.79.Z5.79=5.79−5.430.54=0.666Z_{5.79} = \frac{5.79 -5.43}{0.54} =0.666Z5.79​=0.545.79−5.43​=0.666

STEP 7

STEP 8

Look up the z-score for5.05 in a standard normal distribution table or use a calculator with a normal distribution function.(Z<−0.7037)(Z < -0.7037)(Z<−0.7037)

STEP 9

Calculate the proportion of observations less than5.05.(Z<−.7037)=.2408(Z < -.7037) =.2408(Z<−.7037)=.2408

STEP 10

Look up the z-score for5.79 in a standard normal distribution table or use a calculator with a normal distribution function.(Z<0.6667)(Z <0.6667)(Z<0.6667)

STEP 11

Calculate the proportion of observations less than5.79.(Z<0.6667)=0.7525(Z <0.6667) =0.7525(Z<0.6667)=0.7525The proportion of observations less than5.05 is0.2408 and the proportion of observations less than5.79 is0.7525.

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QuestionFor a distribution $N(5.43,0.54)$ , find the proportion of observations less than 5.05 and 5.79, rounded to four decimal places.

Now, plug in the given values for the first value (5.05), mean and standard deviation to calculate the z-score.
$Z_{5.05} = \frac{5.05 -5.43}{0.54}$

Calculate the z-score for the value.05.
$Z_{.05} = \frac{.05 -.43}{0.54} = -0.7037$

Now, plug in the given values for the second value (5.79), mean and standard deviation to calculate the z-score.
$Z_{5.79} = \frac{5.79 -5.43}{0.54}$

Calculate the z-score for the value5.79.
$Z_{5.79} = \frac{5.79 -5.43}{0.54} =0.666$

Look up the z-score for5.05 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z < -0.7037)$

Calculate the proportion of observations less than5.05.
$(Z < -.7037) =.2408$

Look up the z-score for5.79 in a standard normal distribution table or use a calculator with a normal distribution function.
$(Z <0.6667)$

Calculate the proportion of observations less than5.79.
$(Z <0.6667) =0.7525$ The proportion of observations less than5.05 is0.2408 and the proportion of observations less than5.79 is0.7525.