Math / Data & Statistics

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SAT scores: Assume that in a given year the mean mathematics SAT score was 522, and the standard deviation was 116. A sample of 66 scores is chosen. Use the TI-84 Plus calculator. Español
Part 1 of 5 (a) What is the probability that the sample mean score is less than 509? Round the answer to at least four decimal places.
The probability that the sample mean score is less than 509 is $0.1841^{8}$ .
Correct Answer:
The probability that the sample mean score is less than 509 is 0.1813 .
Part 2 of 5 (b) What is the probability that the sample mean score is between 486 and 525? Round the answer to at least four decimal places.
The probability that the sample mean score is between 486 and 525 is 0.5773
Part: $2 / 5$
Part 3 of 5 (c) Find the $90^{\text {th }}$ percentile of the sample mean. Round the answer to at least two decimal places.
The $90^{\text {th }}$ percentile of the samplemean is $\square$ . part Check Save For Later Submit Assignment @ 2024 MeGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center I Accessibility

Studdy Solution

STEP 1

What is this asking? If we grab 66 random SAT math scores, what's the chance the average is below 509, between 486 and 525, and what's the score that 90% of sample averages will be below? Watch out! Don't mix up the standard deviation of the population with the standard deviation of the sample mean!

STEP 2

1. Calculate the standard error.
2. Calculate the z-scores.
3. Calculate the probabilities.
4. Calculate the 90th percentile.

STEP 3

We know the population standard deviation is $\sigma = 116$ , and our sample size is $n = 66$ .
The standard error (the standard deviation of the sample mean) is calculated by dividing the population standard deviation by the square root of the sample size.
This tells us how spread out the sample means are likely to be.

STEP 4

$\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{66}} \approx 14.26$ So our standard error is approximately 14.26.

STEP 5

The z-score tells us how many standard errors a value is away from the mean.
We'll need to calculate z-scores for 509, 486, and later for the 90th percentile.
The population mean is $\mu = 522$ .

STEP 6

For 509: $z_{509} = \frac{509 - \mu}{\text{Standard Error}} = \frac{509 - 522}{14.26} \approx -0.91$

STEP 7

For 486: $z_{486} = \frac{486 - 522}{14.26} \approx -2.52$

STEP 8

For 525: $z_{525} = \frac{525 - 522}{14.26} \approx 0.21$

STEP 9

Using a z-table or calculator, we can find the probabilities associated with these z-scores.

STEP 10

The probability of a sample mean less than 509 is the area to the left of $z = -0.91$ , which is approximately 0.1813.

STEP 11

The probability of a sample mean between 486 and 525 is the area between $z = -2.52$ and $z = 0.21$ .
This is calculated by subtracting the area to the left of $z = -2.52$ from the area to the left of $z = 0.21$ .
This gives us approximately $0.5834 - 0.0059 =$ 0.5775.

STEP 12

The 90th percentile corresponds to a z-score of approximately 1.28.
We find this using a z-table or calculator by looking for the z-score where the area to the left is 0.90.

STEP 13

Now we can work backwards to find the sample mean corresponding to this z-score: $\text{Sample Mean} = \mu + z \cdot \text{Standard Error}$ $\text{Sample Mean} = 522 + 1.28 \cdot 14.26 \approx 522 + 18.25 \approx 540.25$

STEP 14

(a) The probability that the sample mean score is less than 509 is 0.1813.
(b) The probability that the sample mean score is between 486 and 525 is 0.5775.
(c) The 90th percentile of the sample mean is 540.25.

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Studdy Solution

STEP 1

What is this asking? If we grab 66 random SAT math scores, what's the chance the average is below 509, between 486 and 525, and what's the score that 90% of sample averages will be below? Watch out! Don't mix up the standard deviation of the population with the standard deviation of the sample mean!

STEP 2

1. Calculate the standard error.
2. Calculate the z-scores.
3. Calculate the probabilities.
4. Calculate the 90th percentile.

STEP 3

STEP 4

$\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{66}} \approx 14.26$ So our standard error is approximately 14.26.

STEP 5

The z-score tells us how many standard errors a value is away from the mean.
We'll need to calculate z-scores for 509, 486, and later for the 90th percentile.
The population mean is $\mu = 522$ .

STEP 6

For 509: $z_{509} = \frac{509 - \mu}{\text{Standard Error}} = \frac{509 - 522}{14.26} \approx -0.91$

STEP 7

For 486: $z_{486} = \frac{486 - 522}{14.26} \approx -2.52$

STEP 8

For 525: $z_{525} = \frac{525 - 522}{14.26} \approx 0.21$

STEP 9

Using a z-table or calculator, we can find the probabilities associated with these z-scores.

STEP 10

The probability of a sample mean less than 509 is the area to the left of $z = -0.91$ , which is approximately 0.1813.

STEP 11

The probability of a sample mean between 486 and 525 is the area between $z = -2.52$ and $z = 0.21$ .
This is calculated by subtracting the area to the left of $z = -2.52$ from the area to the left of $z = 0.21$ .
This gives us approximately $0.5834 - 0.0059 =$ 0.5775.

STEP 12

The 90th percentile corresponds to a z-score of approximately 1.28.
We find this using a z-table or calculator by looking for the z-score where the area to the left is 0.90.

STEP 13

Now we can work backwards to find the sample mean corresponding to this z-score: $\text{Sample Mean} = \mu + z \cdot \text{Standard Error}$ $\text{Sample Mean} = 522 + 1.28 \cdot 14.26 \approx 522 + 18.25 \approx 540.25$

STEP 14

(a) The probability that the sample mean score is less than 509 is 0.1813.
(b) The probability that the sample mean score is between 486 and 525 is 0.5775.
(c) The 90th percentile of the sample mean is 540.25.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the standard error.2. Calculate the z-scores.3. Calculate the probabilities.4. Calculate the 90th percentile.

STEP 3

STEP 4

Standard Error=σn=11666≈14.26\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{66}} \approx 14.26Standard Error=n​σ​=66​116​≈14.26 So our **standard error** is approximately **14.26**.

STEP 5

The **z-score** tells us how many standard errors a value is away from the mean.We'll need to calculate z-scores for 509, 486, and later for the 90th percentile.The **population mean** is μ=522\mu = 522μ=522.

STEP 6

For 509: z509=509−μStandard Error=509−52214.26≈−0.91z_{509} = \frac{509 - \mu}{\text{Standard Error}} = \frac{509 - 522}{14.26} \approx -0.91z509​=Standard Error509−μ​=14.26509−522​≈−0.91

STEP 7

For 486: z486=486−52214.26≈−2.52z_{486} = \frac{486 - 522}{14.26} \approx -2.52z486​=14.26486−522​≈−2.52

STEP 8

For 525: z525=525−52214.26≈0.21z_{525} = \frac{525 - 522}{14.26} \approx 0.21z525​=14.26525−522​≈0.21

STEP 9

Using a z-table or calculator, we can find the probabilities associated with these z-scores.

STEP 10

The probability of a sample mean less than 509 is the area to the left of z=−0.91z = -0.91z=−0.91, which is approximately **0.1813**.

STEP 11

STEP 12

The 90th percentile corresponds to a z-score of approximately **1.28**.We find this using a z-table or calculator by looking for the z-score where the area to the left is 0.90.

STEP 13

STEP 14

(a) The probability that the sample mean score is less than 509 is **0.1813**.(b) The probability that the sample mean score is between 486 and 525 is **0.5775**.(c) The 90th percentile of the sample mean is **540.25**.

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1. Calculate the standard error.
2. Calculate the z-scores.
3. Calculate the probabilities.
4. Calculate the 90th percentile.

$\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{66}} \approx 14.26$ So our standard error is approximately 14.26.

The z-score tells us how many standard errors a value is away from the mean.
We'll need to calculate z-scores for 509, 486, and later for the 90th percentile.
The population mean is $\mu = 522$ .

For 509: $z_{509} = \frac{509 - \mu}{\text{Standard Error}} = \frac{509 - 522}{14.26} \approx -0.91$

For 486: $z_{486} = \frac{486 - 522}{14.26} \approx -2.52$

For 525: $z_{525} = \frac{525 - 522}{14.26} \approx 0.21$

The probability of a sample mean less than 509 is the area to the left of $z = -0.91$ , which is approximately 0.1813.

The 90th percentile corresponds to a z-score of approximately 1.28.
We find this using a z-table or calculator by looking for the z-score where the area to the left is 0.90.

(a) The probability that the sample mean score is less than 509 is 0.1813.
(b) The probability that the sample mean score is between 486 and 525 is 0.5775.
(c) The 90th percentile of the sample mean is 540.25.