SAT scores: A college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics SAT score to be 452 . Assume the population standard deviation is $$ \sigma=115 $$. Part: $$ 0 / 4 $$ Part 1 of 4 (a) Construct a $$ 95 \% $$ confidence interval for the mean mathematics SAT score for the entering freshman class. Round the answer to the nearest whole number. A $$ 95 \% $$ confidence interval for the mean mathematics SAT score is (

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To construct a $$ 95\% $$ confidence interval for the mean mathematics SAT score, we can use the formula for the confidence interval when the population standard deviation is known:

$$
\text{CI} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
$$

Where:
- $$\bar{x}$$ is the sample mean,
- $$z$$ is the z-score corresponding to the desired confidence level,
- $$\sigma$$ is the population standard deviation,
- $$n$$ is the sample size.

### Step 1: Identify the known values
- Sample mean ($$\bar{x}$$) = 452
- Population standard deviation ($$\sigma$$) = 115
- Sample size ($$n$$) = 100
- Confidence level = $$95\%$$

### Step 2: Find the z-score for a $$95\%$$ confidence level
For a $$95\%$$ confidence level, the z-score (from the standard normal distribution) is approximately $$1.96$$.

### Step 3: Calculate the standard error (SE)
The standard error (SE) is calculated as follows:

$$
SE = \frac{\sigma}{\sqrt{n}} = \frac{115}{\sqrt{100}} = \frac{115}{10} = 11.5
$$

### Step 4: Calculate the margin of error (ME)
The margin of error (ME) is calculated as follows:

$$
ME = z \cdot SE = 1.96 \cdot 11.5
$$

Now, let's calculate the margin of error.
Calculate the value by following steps:
- step0: Calculate:
  $$1.96\times 11.5$$
- step1: Multiply the numbers:
  $$22.54$$
The margin of error (ME) is approximately $$ 22.54 $$.

### Step 5: Construct the confidence interval
Now we can construct the confidence interval using the sample mean and the margin of error:

$$
\text{CI} = \bar{x} \pm ME = 452 \pm 22.54
$$

Calculating the lower and upper bounds:

- Lower bound: $$ 452 - 22.54 $$
- Upper bound: $$ 452 + 22.54 $$

Let's calculate these bounds.
Calculate the value by following steps:
- step0: Calculate:
  $$452-22.54$$
- step1: Subtract the numbers:
  $$429.46$$
Calculate or simplify the expression $$ 452 + 22.54 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$452+22.54$$
- step1: Add the numbers:
  $$474.54$$
The lower and upper bounds of the confidence interval are:

- Lower bound: $$ 429.46 $$
- Upper bound: $$ 474.54 $$

### Step 6: Round to the nearest whole number
Rounding these values to the nearest whole number gives us:

- Lower bound: $$ 429 $$
- Upper bound: $$ 475 $$

### Final Answer
The $$ 95\% $$ confidence interval for the mean mathematics SAT score is:

$$
(429, 475)
$$
The $$95\%$$ confidence interval for the mean mathematics SAT score is (429, 475).

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