Question

a simple random sample of 90 is drawn from a normally distributed population, and the mean is found to be 138, with a standard deviation of 34. what is the 90% confidence interval for the population mean? use the table below to help you answer the question.

confidence level	90%	95%	99%

remember, the margin of error, me, can be determined using the formula $me = \frac{z \cdot s}{\sqrt{n}}$

• 128.75 to 147.25
• 130.98 to 145.02
• 132.10 to 143.90
• 137.38 to 138.62

Explanation:

Step1: Identify given values

We have: sample size \( n = 90 \), sample mean \( \bar{x}=138 \), standard deviation \( s = 34 \), and for 90% confidence level, \( z^* = 1.645 \).

Step2: Calculate margin of error (ME)

Using the formula \( ME=\frac{z^*\cdot s}{\sqrt{n}} \), substitute the values:
\( ME=\frac{1.645\times34}{\sqrt{90}} \)
First, calculate \( \sqrt{90}\approx9.4868 \)
Then, \( 1.645\times34 = 55.93 \)
Now, \( ME=\frac{55.93}{9.4868}\approx5.90 \) (approximate value, more precise calculation: \( \frac{55.93}{9.4868}\approx5.90 \), actually more accurately: \( 1.645\times34 = 55.93 \), \( \sqrt{90}\approx9.486832981 \), so \( ME=\frac{55.93}{9.486832981}\approx5.90 \) (wait, let's recalculate: 1.64534 = 55.93; 55.93 divided by sqrt(90) ≈ 55.93 / 9.4868 ≈ 5.90? Wait, no, wait 9.48685.9 = 55.97, close. Wait, maybe my approximation is off. Wait, let's do it more accurately:

\( \sqrt{90} = 3\sqrt{10} \approx 9.486832980505138 \)

\( 1.645\times34 = 55.93 \)

\( ME = \frac{55.93}{9.486832980505138} \approx 5.90 \)? Wait, no, 9.48685.9 = 9.48685 + 9.4868*0.9 = 47.434 + 8.53812 = 55.97212, which is close to 55.93, so ME ≈ 5.896, approximately 5.90.

Step3: Calculate confidence interval

The confidence interval is \( \bar{x} - ME \) to \( \bar{x} + ME \)

Lower bound: \( 138 - 5.90 = 132.10 \)

Upper bound: \( 138 + 5.90 = 143.90 \)

Wait, let's check with more precise calculation:

ME = (1.645 * 34)/sqrt(90) = (55.93)/9.48683298 ≈ 5.896 ≈ 5.90

So 138 - 5.896 ≈ 132.104, 138 + 5.896 ≈ 143.896, which rounds to 132.10 to 143.90.

Answer:

132.10 to 143.90