3. 1800 freshmen were admitted to greenville tech this fall. the scores on the sat were distributed normally with a mean of 520 and a standard deviation of 90.
label the mean and three standard deviations from the mean.
what percent of students scored between 340 and 700?
what percent of students scored between 430 and 700?
what percent of students scored below 430?
what percent of students scored above 700?
how many students would you expect scored below 520?
how many students would you expect scored above 610?

Question

Accepted Answer

### Part 1: Label the mean and three standard deviations from the mean
# Explanation:
## Step 1: Identify the mean and standard deviation
The mean ($\mu$) is 520, and the standard deviation ($\sigma$) is 90.

## Step 2: Calculate values for three standard deviations below and above the mean
- One standard deviation below the mean: $\mu - \sigma = 520 - 90 = 430$
- Two standard deviations below the mean: $\mu - 2\sigma = 520 - 2	imes90 = 520 - 180 = 340$
- Three standard deviations below the mean: $\mu - 3\sigma = 520 - 3	imes90 = 520 - 270 = 250$
- One standard deviation above the mean: $\mu + \sigma = 520 + 90 = 610$
- Two standard deviations above the mean: $\mu + 2\sigma = 520 + 2	imes90 = 520 + 180 = 700$
- Three standard deviations above the mean: $\mu + 3\sigma = 520 + 3	imes90 = 520 + 270 = 790$

So the labels from left to right (assuming the curve is centered at the mean) would be: 250 (3$\sigma$ below), 340 (2$\sigma$ below), 430 (1$\sigma$ below), 520 (mean), 610 (1$\sigma$ above), 700 (2$\sigma$ above), 790 (3$\sigma$ above).

### Part 2: What percent of students scored between 340 and 700?
# Explanation:
## Step 1: Identify the number of standard deviations from the mean
340 is $\mu - 2\sigma$ (since $520 - 2	imes90 = 340$) and 700 is $\mu + 2\sigma$ (since $520 + 2	imes90 = 700$).

## Step 2: Use the empirical rule for normal distribution
The empirical rule states that for a normal distribution:
- Approximately 68% of the data lies within $\mu \pm \sigma$
- Approximately 95% of the data lies within $\mu \pm 2\sigma$
- Approximately 99.7% of the data lies within $\mu \pm 3\sigma$

Since 340 is $\mu - 2\sigma$ and 700 is $\mu + 2\sigma$, the percentage of data between them is approximately 95%.

# Answer: 95%

### Part 3: What percent of students scored between 430 and 700?
# Explanation:
## Step 1: Identify the number of standard deviations from the mean
430 is $\mu - \sigma$ (since $520 - 90 = 430$) and 700 is $\mu + 2\sigma$ (since $520 + 2	imes90 = 700$).

## Step 2: Calculate the percentage using the empirical rule
- Percentage within $\mu - \sigma$ to $\mu + \sigma$: 68%
- Percentage within $\mu + \sigma$ to $\mu + 2\sigma$: $\frac{95\% - 68\%}{2} = 13.5\%$ (since the total within $\mu \pm 2\sigma$ is 95%, and we already have 68% within $\mu \pm \sigma$, the remaining is split equally on both sides)

So the percentage between $\mu - \sigma$ (430) and $\mu + 2\sigma$ (700) is $68\% + 13.5\% = 81.5\%$

# Answer: 81.5%

### Part 4: What percent of students scored below 430?
# Explanation:
## Step 1: Identify the number of standard deviations from the mean
430 is $\mu - \sigma$ (since $520 - 90 = 430$).

## Step 2: Use the empirical rule
The total percentage of data below $\mu$ is 50%. The percentage of data between $\mu - \sigma$ and $\mu$ is $\frac{68\%}{2} = 34\%$ (since 68% is within $\mu \pm \sigma$, split equally on both sides of the mean).

So the percentage of data below $\mu - \sigma$ (430) is $50\% - 34\% = 16\%$ (because 50% is below the mean, and 34% is between the mean and $\mu - \sigma$, so the remaining below $\mu - \sigma$ is 50% - 34% = 16%).

# Answer: 16%

### Part 5: What percent of students scored above 700?
# Explanation:
## Step 1: Identify the number of standard deviations from the mean
700 is $\mu + 2\sigma$ (since $520 + 2	imes90 = 700$).

## Step 2: Use the empirical rule
The total percentage of data above $\mu$ is 50%. The percentage of data between $\mu$ and $\mu + 2\sigma$ is $\frac{95\%}{2} = 47.5\%$ (since 95% is within $\mu \pm 2\sigma$, split equally on both sides of the mean).

So the percentage of data above $\mu + 2\sigma$ (700) is $50\% - 47.5\% = 2.5\%$

# Answer: 2.5%

### Part 6: How many students would you expect scored below 520?
# Explanation:
## Step 1: Recognize the mean in a normal distribution
In a normal distribution, the mean is the center, so 50% of the data is below the mean and 50% is above.

## Step 2: Calculate the number of students
Total number of students is 1800. So the number of students below 520 is $0.5	imes1800 = 900$

# Answer: 900

### Part 7: How many students would you expect scored above 610?
# Explanation:
## Step 1: Identify the number of standard deviations from the mean
610 is $\mu + \sigma$ (since $520 + 90 = 610$).

## Step 2: Use the empirical rule to find the percentage above $\mu + \sigma$
From the empirical rule, we know that 68% of the data is within $\mu \pm \sigma$. So the percentage of data above $\mu + \sigma$ is $\frac{100\% - 68\%}{2} = 16\%$ (because the remaining 32% is split equally above $\mu + \sigma$ and below $\mu - \sigma$).

## Step 3: Calculate the number of students
Number of students above 610 is $0.16	imes1800 = 288$

# Answer: 288

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QUESTION IMAGE

Question

Explanation:

Part 1: Label the mean and three standard deviations from the mean

Step 1: Identify the mean and standard deviation

Step 2: Calculate values for three standard deviations below and above the mean

Part 2: What percent of students scored between 340 and 700?

Step 1: Identify the number of standard deviations from the mean

Step 2: Use the empirical rule for normal distribution

Step 1: Identify the number of standard deviations from the mean

Step 2: Calculate the percentage using the empirical rule

Answer:

Part 3: What percent of students scored between 430 and 700?