Question

A sample of 64
bolts has a mean diameter of 2.5
cm.

The standard deviation of the sample is 0.4
cm.

Using z=1.96,
calculate a 95%
confidence interval for the true mean diameter.

Drag and drop the appropriate values into the confidence interval.

Click Here for a Hint

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Response area ≤μ≤
Response area
: 2.304
:: 2.402
:: 2.696
: 2.598
:: 2.998

Answer 1

To calculate the 95% confidence interval for the true mean diameter using the provided data, we can use the formula for the confidence interval:

\[
\text{Confidence Interval} = \bar{x} \pm z \times \left(\frac{s}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) = sample mean = 2.5 cm
- \(z\) = z-score for 95% confidence level = 1.96
- \(s\) = sample standard deviation = 0.4 cm
- \(n\) = sample size = 64

First, we need to calculate the standard error (SE):

\[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{0.4}{\sqrt{64}} = \frac{0.4}{8} = 0.05
\]

Next, we can calculate the margin of error (ME):

\[
\text{ME} = z \times \text{SE} = 1.96 \times 0.05 = 0.098
\]

Now, we can determine the confidence interval:

\[
\text{Lower limit} = \bar{x} - \text{ME} = 2.5 - 0.098 = 2.402
\]
\[
\text{Upper limit} = \bar{x} + \text{ME} = 2.5 + 0.098 = 2.598
\]

Thus, the 95% confidence interval for the true mean diameter is:

\[
2.402 \leq \mu \leq 2.598
\]

The values to drag and drop into the response areas are:

- Lower limit: **2.402**
- Upper limit: **2.598**

So the correct inputs would be:

\(2.402 \text{ (lower limit)} \leq \mu \leq 2.598 \text{ (upper limit)}\)