The state test scores for 12 randomly selected high school seniors are shown on the right. Complete parts​ (a) through​ (c) below.
 Assume the population is normally distributed.
1430
1228
981
 
696
723
832
720
746
542
628
1449
949
Question content area bottom
Part 1
​(a) Find the sample mean.
x overbarequals
  
910.3 ​(Round to one decimal place as​ needed.)
Part 2
​(b) Find the sample standard deviation.
sequals
  
306.5 ​(Round to one decimal place as​ needed.)
Part 3
​(c) Construct a 90​% confidence interval for the population mean mu.
A 90​% confidence interval for the population mean is ​(
  
enter your response here​,
  
enter your response here​). 
​(Round to one decimal place as​ needed.)

Let's work through each part of the problem step by step.

### Part (a): Sample Mean
You already have the sample mean \( \bar{x} = 910.3 \) (rounded to one decimal place), so we can move on.

### Part (b): Sample Standard Deviation
The sample standard deviation \( s = 306.5 \) has also been provided.

### Part (c): Construct a 90% Confidence Interval for the Population Mean

To calculate the confidence interval, we will use the following formula for the confidence interval of the mean when the population standard deviation is unknown:

\[
\text{CI} = \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}
\]

Where:
- \( \bar{x} = 910.3 \) is the sample mean,
- \( s = 306.5 \) is the sample standard deviation,
- \( n = 12 \) is the sample size,
- \( t_{\alpha/2} \) is the critical value from the t-distribution for a 90% confidence level with \( n-1 = 11 \) degrees of freedom.

#### Step 1: Find the critical value \( t_{\alpha/2} \)
For a 90% confidence interval and 11 degrees of freedom, the critical t-value can be found in a t-distribution table or using technology. The value is approximately:

\[
t_{\alpha/2} = 1.796
\]

#### Step 2: Calculate the margin of error (MOE)
The margin of error is given by:

\[
\text{MOE} = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 1.796 \cdot \frac{306.5}{\sqrt{12}}
\]

Let me calculate the margin of error for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

For now, you can calculate the margin of error manually using the formula:

\[
\text{MOE} = 1.796 \cdot \frac{306.5}{\sqrt{12}} \approx 1.796 \cdot 88.45 \approx 158.8
\]

So, the margin of error is approximately 158.8.

### Step 3: Calculate the Confidence Interval
To find the confidence interval:

\[
\text{Lower bound} = \bar{x} - \text{MOE} = 910.3 - 158.8 = 751.5
\]
\[
\text{Upper bound} = \bar{x} + \text{MOE} = 910.3 + 158.8 = 1069.1
\]

Thus, the 90% confidence interval for the population mean is approximately \( (751.5, 1069.1) \).

Would you like further details or have any questions?

Here are 5 related questions for deeper exploration:
1. How does the sample size affect the width of the confidence interval?
2. Why do we use the t-distribution instead of the z-distribution for this problem?
3. What happens to the confidence interval if the sample standard deviation increases?
4. How would you interpret this confidence interval in practical terms?
5. How would a 95% confidence interval differ from a 90% one?

**Tip:** As the sample size increases, the t-distribution approaches the z-distribution, leading to narrower confidence intervals.

The state test scores for 12 randomly selected high school seniors are shown on the right. Complete parts (a) through (c) below. Assume the population is normally distributed. Scores: 1430, 1228, 981, 696, 723, 832, 720, 746, 542, 628, 1449, 949. (a) Find the sample mean. (b) Find the sample standard deviation. (c) Construct a 90% confidence interval for the population mean.

Learn how to find the sample mean, sample standard deviation, and construct a 90% confidence interval for the population mean using a sample of 12 test scores. This problem involves key statistical concepts like the t-distribution and sample data analysis, suitable for high school seniors or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation