2. 2000 freshmen at state university took a biology test. the scores were distributed normally with a mean of 70 and a standard deviation of 5.
label the mean and three standard deviations from the mean.
what percentage of scores are between scores 65 and 75?
what percentage of scores are between scores 60 and 85?
what percentage of scores is less than a score of 55?
what percentage of scores is greater than a score of 80?
approximately how many biology students scored between 60 and 70?
approximately how many biology students scored between 55 and 60?

Question

Accepted Answer

### 1. Label the mean and three standard deviations from the mean
# Explanation:
## Step1: Identify the mean
The mean ($\mu$) is given as 70. So the center of the normal curve is at 70.
## Step2: Calculate one standard deviation from the mean
The standard deviation ($\sigma$) is 5.
- One standard deviation below the mean: $\mu - \sigma = 70 - 5 = 65$
- One standard deviation above the mean: $\mu + \sigma = 70 + 5 = 75$
## Step3: Calculate two standard deviations from the mean
- Two standard deviations below the mean: $\mu - 2\sigma = 70 - 2	imes5 = 70 - 10 = 60$
- Two standard deviations above the mean: $\mu + 2\sigma = 70 + 2	imes5 = 70 + 10 = 80$
## Step4: Calculate three standard deviations from the mean
- Three standard deviations below the mean: $\mu - 3\sigma = 70 - 3	imes5 = 70 - 15 = 55$
- Three standard deviations above the mean: $\mu + 3\sigma = 70 + 3	imes5 = 70 + 15 = 85$

So the labels from left to right (assuming the curve is symmetric around 70) would be: 55, 60, 65, 70, 75, 80, 85.

### 2. What percentage of scores are between 65 and 75?
# Explanation:
## Step1: Recognize the range
65 is $\mu - \sigma$ and 75 is $\mu + \sigma$. For a normal distribution, the percentage of data between $\mu - \sigma$ and $\mu + \sigma$ is approximately 68% (Empirical Rule).
# Answer:
68%

### 3. What percentage of scores are between 60 and 85?
# Explanation:
## Step1: Identify the number of standard deviations
- 60 is $\mu - 2\sigma$ (since $70 - 2	imes5 = 60$)
- 85 is $\mu + 3\sigma$ (since $70 + 3	imes5 = 85$)
## Step2: Use the Empirical Rule
- Percentage between $\mu - 2\sigma$ and $\mu + 2\sigma$ is 95%
- Percentage between $\mu + 2\sigma$ and $\mu + 3\sigma$ is approximately $\frac{99.7\% - 95\%}{2}= 2.35\%$
- So total percentage between 60 ($\mu - 2\sigma$) and 85 ($\mu + 3\sigma$) is $95\% + 2.35\% = 97.35\%$ (or approximately 97.5% if we consider the empirical rule approximations more roughly)
# Answer:
Approximately 97.5% (or 97.35%)

### 4. What percentage of scores is less than a score of 55?
# Explanation:
## Step1: Identify the position of 55
55 is $\mu - 3\sigma$ (since $70 - 3	imes5 = 55$)
## Step2: Use the Empirical Rule
For a normal distribution, the percentage of data less than $\mu - 3\sigma$ is approximately $\frac{100\% - 99.7\%}{2}= 0.15\%$
# Answer:
0.15%

### 5. What percentage of scores is greater than a score of 80?
# Explanation:
## Step1: Identify the position of 80
80 is $\mu + 2\sigma$ (since $70 + 2	imes5 = 80$)
## Step2: Use the Empirical Rule
The percentage of data greater than $\mu + 2\sigma$ is $\frac{100\% - 95\%}{2}= 2.5\%$
# Answer:
2.5%

### 6. Approximately how many biology students scored between 60 and 70?
# Explanation:
## Step1: Find the percentage of scores between 60 and 70
- 60 is $\mu - 2\sigma$ and 70 is the mean ($\mu$)
- The percentage of data between $\mu - 2\sigma$ and $\mu$ is half of the percentage between $\mu - 2\sigma$ and $\mu + 2\sigma$. Since the percentage between $\mu - 2\sigma$ and $\mu + 2\sigma$ is 95%, the percentage between $\mu - 2\sigma$ and $\mu$ is $\frac{95\%}{2}= 47.5\%$
## Step2: Calculate the number of students
Total number of students is 2000. So the number of students with scores between 60 and 70 is $2000	imes47.5\% = 2000	imes0.475 = 950$
# Answer:
950

### 7. Approximately how many biology students scored between 55 and 60?
# Explanation:
## Step1: Find the percentage of scores between 55 and 60
- 55 is $\mu - 3\sigma$ and 60 is $\mu - 2\sigma$
- The percentage of data between $\mu - 3\sigma$ and $\mu - 2\sigma$ is $\frac{99.7\% - 95\%}{2}= 2.35\%$ (since the percentage between $\mu - 3\sigma$ and $\mu + 3\sigma$ is 99.7% and between $\mu - 2\sigma$ and $\mu + 2\sigma$ is 95%, so the area between $\mu - 3\sigma$ and $\mu - 2\sigma$ is half of the difference)
## Step2: Calculate the number of students
Total number of students is 2000. So the number of students with scores between 55 and 60 is $2000	imes2.35\% = 2000	imes0.0235 = 47$ (or approximately 50 if we use more approximate values)
# Answer:
Approximately 47 (or 50)

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QUESTION IMAGE

Question

Explanation:

1. Label the mean and three standard deviations from the mean

Step1: Identify the mean

Step2: Calculate one standard deviation from the mean

Step3: Calculate two standard deviations from the mean

Step4: Calculate three standard deviations from the mean

2. What percentage of scores are between 65 and 75?

Step1: Recognize the range

Step1: Identify the number of standard deviations

Step2: Use the Empirical Rule

Answer:

3. What percentage of scores are between 60 and 85?