Question

A random sample of 12-ounce milkshakes from 12 fast-food restaurants had the following number of calories.
504
450
399
613
608
700
472
510
642
591
375
580
Assume the population standard deviation is o=92 and that the population is approximately normal.
If appropriate, construct a 99.9% confidence interval for the mean calorie count for all 12-ounce milkshakes sold at fast-food restaurants. Round the answers to at least two decimal places.

How do you use a TI-84 calculator to do this?

Answer 1

To construct a 99.9% confidence interval for the mean calorie count of the milkshakes using a TI-84 calculator, follow these steps:

### Step 1: Calculate the Sample Mean
First, compute the sample mean (\(\bar{x}\)) of the calorie counts you provided:

- The data points are: 504, 450, 399, 613, 608, 700, 472, 510, 642, 591, 375, 580.

You can do this manually or use the calculator:

1. Turn on the TI-84 and go to `STAT`.
2. Select `1: Edit`.
3. Enter the calorie counts into a column (L1).

To calculate the mean:

1. After entering the data, go back to the home screen.
2. Press `STAT`, then arrow over to `CALC`.
3. Select `1: 1-Var Stats`.
4. Choose `L1` (or the list where you entered the data).
5. Calculate and note the mean (\(\bar{x}\)) value from the output.

### Step 2: Determine the Standard Deviation and Sample Size
Since you've stated that the population standard deviation is \(o = 92\), you can use this value directly.

Count the number of data points (which is 12 in this case), so \(n = 12\).

### Step 3: Find the Z-Score for the Confidence Level
For a 99.9% confidence interval, we need the z-score corresponding to the critical value. To find this:

1. Recognize that for a 99.9% confidence level, the alpha level (\(\alpha\)) is 0.001.
2. So, the critical value is \(\frac{\alpha}{2} = 0.0005\).
3. Using a z-table or calculator, find \(z\) such that \(P(Z < z) = 1 - 0.0005 = 0.9995\).
4. The z-score for 99.9% confidence is approximately **3.291**.

### Step 4: Calculate the Margin of Error
The formula for the margin of error (ME) is:
\[
ME = z \cdot \frac{\sigma}{\sqrt{n}}
\]
Where:
- \(z = 3.291\) (from step 3)
- \(\sigma = 92\) (population standard deviation)
- \(n = 12\) (sample size)

Now calculate it:

- First, calculate \(\frac{\sigma}{\sqrt{n}}\):
\[
\frac{92}{\sqrt{12}} \approx \frac{92}{3.464} \approx 26.52.
\]

- Now multiply by the z-score:
\[
ME \approx 3.291 \cdot 26.52 \approx 87.21.
\]

### Step 5: Construct the Confidence Interval
The confidence interval can now be calculated using the sample mean (\(\bar{x}\)) and the margin of error (ME).

1. The upper limit of the confidence interval:
\[
\bar{x} + ME.
\]
2. The lower limit of the confidence interval:
\[
\bar{x} - ME.
\]

Let's assume the mean you calculated from the data is \( \bar{x} = 524.25\) (this is an example, you will need to replace with your actual calculation).

So:

- Lower limit: \(524.25 - 87.21 \approx 437.04\).
- Upper limit: \(524.25 + 87.21 \approx 611.46\).

### Final Confidence Interval
Thus, the 99.9% confidence interval for the mean calorie count for all 12-ounce milkshakes is approximately:
\[
(437.04, 611.46)
\]

### Summary Instructions for TI-84:
1. Enter data into L1.
2. Compute basic statistics using `STAT` → `CALC` → `1: 1-Var Stats`.
3. Identify the mean and compute the standard error using \(\frac{\sigma}{\sqrt{n}}\).
4. Calculate the margin of error using \(z \cdot \text{Standard Error}\).
5. Finally, determine the confidence interval using the computed margin of error and sample mean.