and the the questions below given that the mean score in a math quiz is 50 standard deviation is 8 Round your answers in percent to the nearest hundredths (like .40346 is 40.35% Round your answers for of people to the nearest whole number (like 8.32 is 8)Do not use What is the area under the curve between 42 and 66? square What is the probability of having a value less than 58? square If there are 48 students who took the quiz, how many students scored greater than 66? square What is the area under the curve between 45 and 60? square If there are 30 students took the quiz, approximately how many students are expected to get less than 65? square

## Step 1: Calculate the z-scores for 42 and 66.<br />### The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For $x = 42$, $z = \frac{42 - 50}{8} = -1$. For $x = 66$, $z = \frac{66 - 50}{8} = 2$.<br /><br />## Step 2: Find the area between z = -1 and z = 2.<br />### Using a z-table or calculator, the area to the left of $z = 2$ is approximately 0.9772, and the area to the left of $z = -1$ is approximately 0.1587. The area between these two z-scores is $0.9772 - 0.1587 = 0.8185$.<br /><br />## Step 3: Calculate the probability of a value less than 58.<br />### The z-score for $x = 58$ is $z = \frac{58 - 50}{8} = 1$. The area to the left of $z = 1$ is approximately 0.8413.<br /><br />## Step 4: Calculate the number of students scoring greater than 66.<br />### From Step 2, the area to the left of $z = 2$ (corresponding to $x = 66$) is 0.9772. The area to the right is $1 - 0.9772 = 0.0228$. With 48 students, the number scoring greater than 66 is $48 \times 0.0228 \approx 1.0944 \approx 1$.<br /><br />## Step 5: Calculate the area under the curve between 45 and 60.<br />### The z-score for $x = 45$ is $z = \frac{45 - 50}{8} = -0.625$. The z-score for $x = 60$ is $z = \frac{60 - 50}{8} = 1.25$. The area to the left of $z = 1.25$ is approximately 0.8944, and the area to the left of $z = -0.625$ is approximately 0.2660 (interpolating or using a calculator). The area between these z-scores is $0.8944 - 0.2660 = 0.6284$.<br /><br />## Step 6: Calculate the number of students scoring less than 65.<br />### The z-score for $x = 65$ is $z = \frac{65 - 50}{8} = 1.875$. The area to the left of $z = 1.875$ is approximately 0.9696. With 30 students, the number scoring less than 65 is $30 \times 0.9696 \approx 29.088 \approx 29$.