Christine is currently taking a college astronomy class and the instructor often gives quizzes. On the past seve quizzes, Christine got the scores shown below. Find the standard deviation, rounding to one more decimal pla than is present in the original data. 50153127114271 (1 point) o 20.9 437.6 31 8715.6

Question

UpStudy AI · Accepted Answer

To find the standard deviation of Christine's quiz scores, we will follow these steps:

1. **List the scores**: The scores are 50, 153, 127, 114, 271.
2. **Calculate the mean (average)**: The mean is calculated by summing all the scores and dividing by the number of scores.
3. **Calculate the variance**: The variance is the average of the squared differences from the mean.
4. **Calculate the standard deviation**: The standard deviation is the square root of the variance.
5. **Round the result**: Finally, we will round the standard deviation to one more decimal place than is present in the original data.

Let's perform these calculations step by step.

### Step 1: List the scores
The scores are:
- $$ x_1 = 50 $$
- $$ x_2 = 153 $$
- $$ x_3 = 127 $$
- $$ x_4 = 114 $$
- $$ x_5 = 271 $$

### Step 2: Calculate the mean
The mean $$ \mu $$ is calculated as follows:
$$
\mu = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5}
$$

### Step 3: Calculate the variance
The variance $$ \sigma^2 $$ is calculated using the formula:
$$
\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + (x_3 - \mu)^2 + (x_4 - \mu)^2 + (x_5 - \mu)^2}{5}
$$

### Step 4: Calculate the standard deviation
The standard deviation $$ \sigma $$ is the square root of the variance:
$$
\sigma = \sqrt{\sigma^2}
$$

### Step 5: Round the result
We will round the standard deviation to one more decimal place than is present in the original data.

Now, let's perform the calculations.

First, we will calculate the mean. Then, we will use it to find the variance and standard deviation.

Let's proceed with the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(50+153+127+114+271\right)}{5}$$
- step1: Remove the parentheses:
  $$\frac{50+153+127+114+271}{5}$$
- step2: Add the numbers:
  $$\frac{715}{5}$$
- step3: Reduce the numbers:
  $$\frac{143}{1}$$
- step4: Calculate:
  $$143$$
Calculate or simplify the expression $$ \sqrt(((50 - 143)^2 + (153 - 143)^2 + (127 - 143)^2 + (114 - 143)^2 + (271 - 143)^2) / 5) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{\left(\left(50-143\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}\right)}{5}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{\left(50-143\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}}$$
- step2: Subtract the numbers:
  $$\sqrt{\frac{\left(-93\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}}$$
- step3: Subtract the numbers:
  $$\sqrt{\frac{\left(-93\right)^{2}+10^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}}$$
- step4: Subtract the numbers:
  $$\sqrt{\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}}$$
- step5: Subtract the numbers:
  $$\sqrt{\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(-29\right)^{2}+\left(271-143\right)^{2}}{5}}$$
- step6: Subtract the numbers:
  $$\sqrt{\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(-29\right)^{2}+128^{2}}{5}}$$
- step7: Add the numbers:
  $$\sqrt{\frac{26230}{5}}$$
- step8: Divide the terms:
  $$\sqrt{5246}$$
Calculate or simplify the expression $$ ((50 - 143)^2 + (153 - 143)^2 + (127 - 143)^2 + (114 - 143)^2 + (271 - 143)^2) / 5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(\left(50-143\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}\right)}{5}$$
- step1: Remove the parentheses:
  $$\frac{\left(50-143\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}$$
- step2: Subtract the numbers:
  $$\frac{\left(-93\right)^{2}+\left(153-143\right)^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}$$
- step3: Subtract the numbers:
  $$\frac{\left(-93\right)^{2}+10^{2}+\left(127-143\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}$$
- step4: Subtract the numbers:
  $$\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(114-143\right)^{2}+\left(271-143\right)^{2}}{5}$$
- step5: Subtract the numbers:
  $$\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(-29\right)^{2}+\left(271-143\right)^{2}}{5}$$
- step6: Subtract the numbers:
  $$\frac{\left(-93\right)^{2}+10^{2}+\left(-16\right)^{2}+\left(-29\right)^{2}+128^{2}}{5}$$
- step7: Add the numbers:
  $$\frac{26230}{5}$$
- step8: Reduce the numbers:
  $$\frac{5246}{1}$$
- step9: Calculate:
  $$5246$$
Round the number $$ 72.429276 $$ into 1 decimal place.
Round the number by following steps:
- step0: Round $$72.429276$$ to $$1$$ decimal places using standard rounding method:
  $$72.429276$$
- step1: Round $$72.429276$$ to $$1$$ decimal places using standard rounding method$$:$$
  $$72.4$$
Let's summarize the calculations step by step:

1. **Scores**: The scores are 50, 153, 127, 114, and 271.
2. **Mean Calculation**:
   $$
   \mu = \frac{50 + 153 + 127 + 114 + 271}{5} = 143
   $$

3. **Variance Calculation**:
   $$
   \sigma^2 = \frac{(50 - 143)^2 + (153 - 143)^2 + (127 - 143)^2 + (114 - 143)^2 + (271 - 143)^2}{5} = \frac{5246}{5} = 1049.2
   $$

4. **Standard Deviation Calculation**:
   $$
   \sigma = \sqrt{1049.2} \approx 72.429276
   $$

5. **Rounding**: Rounding to one more decimal place than the original data (which has no decimal places), we get:
   $$
   \sigma \approx 72.4
   $$

Thus, the standard deviation of Christine's quiz scores is **72.4**.
The standard deviation of Christine's quiz scores is **72.4**.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Statistics Questions