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A simple random sample of 60 is drawn from a normally distributed population. The mean is found to be 28, with a standard deviation of 5. Which of the following values is within the [tex][tex]$95\%$[/tex][/tex] confidence interval (z-score [tex][tex]$=1.96)$[/tex][/tex] for the population mean?

Remember, the margin of error, [tex][tex]$ME$[/tex][/tex], can be determined using the formula [tex][tex]$ME=\frac{z \cdot s}{\sqrt{n}}$[/tex][/tex].

A. The value of 26, because it's not greater than 26.7 and less than 29.3.
B. The value of 27, because it's greater than 26.7 and less than 29.3.
C. The value of 32, because it's greater than 23 and less than 33.
D. The value of 34, because it's not greater than 23 and less than 33.



Answer :

GinnyAnswer
To determine which of the given values falls within the 95% confidence interval for the population mean, let's carefully follow these steps:

Step 1: Understand the Given Data
- Sample Size ([tex]\(n\)[/tex]): 60
- Sample Mean ([tex]\(\bar{x}\)[/tex]): 28
- Sample Standard Deviation ([tex]\(s\)[/tex]): 5
- [tex]\(z\)[/tex]-score for 95% confidence interval: 1.96

Step 2: Calculate the Margin of Error (ME)
The formula for the margin of error is:
[tex]\[ ME = \frac{z \cdot s}{\sqrt{n}} \][/tex]

Plugging in the given values:
[tex]\[ ME = \frac{1.96 \cdot 5}{\sqrt{60}} \][/tex]

Given that we have pre-calculated the values, we find:
[tex]\[ ME \approx 1.2652 \][/tex]

Step 3: Calculate the Confidence Interval
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean.

- Lower Bound:
[tex]\[ \text{Lower Bound} = \bar{x} - ME = 28 - 1.2652 \approx 26.735 \][/tex]

- Upper Bound:
[tex]\[ \text{Upper Bound} = \bar{x} + ME = 28 + 1.2652 \approx 29.265 \][/tex]

So, the 95% confidence interval is approximately (26.735, 29.265).

Step 4: Check the Given Values Against the Confidence Interval
We need to determine if each value is within the derived confidence interval (26.735, 29.265):

- Value 26:
[tex]\[ 26 < 26.735 \quad \text{(Not within the interval)} \][/tex]

- Value 27:
[tex]\[ 26.735 \leq 27 \leq 29.265 \quad \text{(Within the interval)} \][/tex]

- Value 32:
[tex]\[ 32 > 29.265 \quad \text{(Not within the interval)} \][/tex]

- Value 34:
[tex]\[ 34 > 29.265 \quad \text{(Not within the interval)} \][/tex]

Conclusion:
The only value within the 95% confidence interval is 27. Therefore, the correct interpretation is:

- The value of 27 is within the confidence interval because it is greater than 26.735 and less than 29.265.

Hence, the correct answer is:
The value of 27, because it is greater than 26.7 and less than 29.3.