A random sample of 12-ounce milkshakes from 12 fast-food restaurants had the following number of calories.

Question

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.