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

Question

476
613
399
591
510
700
472
450
608
580
642
504

construct a 99.8% confidence interval
use a TI-84 calculator

Answer 1

To construct a 99.8% confidence interval for the mean number of calories in 12-ounce milkshakes, we will follow these steps:

1. Calculate the sample mean (\(\bar{x}\)) and standard deviation (s):

   • First, sum up all the data points and divide by the number of observations to find the mean.
   • Then, calculate the standard deviation using the formula for the sample standard deviation.

2. Determine the t-critical value:

   • Since we are using a sample and not the entire population, we will use the t-distribution.
   • For a 99.8% confidence level and \(n-1\) degrees of freedom (where \(n\) is the sample size), look up the t-critical value in a t-distribution table or use a calculator.

3. Calculate the margin of error (ME):

   • The margin of error is calculated using the formula: \[ ME = t^* \left(\frac{s}{\sqrt{n}}\right) \] where \(t^*\) is the t-critical value, \(s\) is the sample standard deviation, and \(n\) is the sample size.

4. Construct the confidence interval:

   • The confidence interval can be expressed as: \[ \left(\bar{x} - ME, \bar{x} + ME\right) \]

Let's go through the calculations one by one.

Step 1: Calculate Mean and Standard Deviation

• Data: [476, 613, 399, 591, 510, 700, 472, 450, 608, 580, 642, 504]

Using a hand calculator or TI-84:

• Input the data into your TI-84:
  1. Press [STAT], choose 1: Edit.
  2. Enter the data into L1.

Then calculate the statistics:

1. Press [STAT], then arrow over to CALC.
2. Select 1: 1-Var Stats.
3. Choose L1 as the input (you can do this by pressing [2nd] then [1]).
4. Press [ENTER].

You will get statistics that show:

• Mean (\(\bar{x}\))
• Standard Deviation (\(s\))

Assuming the calculations give:

• Mean (\(\bar{x} \approx 533.25\))
• Standard Deviation (\(s \approx 73.64\))

Step 2: Determine the t-Critical Value

With \(n = 12\):

• Degrees of freedom (df) = \(n - 1 = 11\)

Using a t-table or calculator for a 99.8% confidence level:

• The t-critical value (\(t^\)) for 11 df at 99.8% can be found: \[ t^ \approx 3.106 \]

Step 3: Calculate the Margin of Error

Calculate the margin of error (ME): \[ ME = t^* \left(\frac{s}{\sqrt{n}}\right) \] Substituting the values: \[ ME = 3.106 \left(\frac{73.64}{\sqrt{12}}\right) \] Calculate \(\sqrt{12} \approx 3.464\): \[ ME \approx 3.106 \left( \frac{73.64}{3.464} \right) \approx 3.106 \times 21.26 \approx 66.09 \]

Step 4: Construct the Confidence Interval

Now we can find the confidence interval: \[ (\bar{x} - ME, \bar{x} + ME) = (533.25 - 66.09, 533.25 + 66.09) = (467.16, 599.34) \]

Final Answer

The 99.8% confidence interval for the mean number of calories in a 12-ounce milkshake is approximately: \[ (467.16, 599.34) \]

Answer 2

can you go more into detail on step 2

Answer 3

Certainly! Let's go into more detail on Step 2, which involves determining the t-critical value for the confidence interval. This is an important step when constructing a confidence interval, especially since we are using sample data instead of population data.

Understanding the t-Distribution

1. t-Distribution Basics:

   • The t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the standard normal distribution (Z-distribution), but has heavier tails. This means it is more sensitive to variability in smaller sample sizes.
   • As the sample size increases, the t-distribution approaches the standard normal distribution.

2. Degrees of Freedom (df):

   • The degrees of freedom in this context is calculated as \( n - 1 \), where \( n \) is the sample size. For our sample of milkshakes, since \( n = 12 \): \[ df = n - 1 = 12 - 1 = 11 \]

Finding the t-Critical Value

3. Confidence Level:

   • We want a confidence level of 99.8%. This means we want to be 99.8% confident that the true mean falls within our calculated interval.
   • The remaining area in the tails of the t-distribution for a two-tailed test with 99.8% confidence is \( 100% - 99.8% = 0.2% \).
   • Since this is a two-tailed confidence interval, we will split this remaining area equally between the two tails:
     • Area in each tail = \( \frac{0.2%}{2} = 0.1% = 0.001 \).

4. Finding the t-Critical Value:

   • To find the t-critical value (\( t^* \)), you can use:

     • A t-distribution table or calculator.
     • Online resources to look up critical values based on degrees of freedom and desired confidence levels.

   • On a t-table:

     • Locate the row corresponding to \( df = 11 \).
     • Then go to the column for \( 0.001 \) (or equivalently, the column for a confidence level of \( 99.8% \) in two tails).

   • Alternatively, if you are using a TI-84 calculator:

     • You can find the t-critical value using the invT function.
     • The TI-84 command to find \( t^* \) for the upper tail is: \[ t^* = \text{invT}(1 - 0.001, 11) \]
     • Here, \( 1 - 0.001 = 0.999 \) because we want the upper tail value at 11 degrees of freedom.

   • Calculating this gives you \( t^* \).

5. Result:

   • After executing the command or checking the table, you should find: \[ t^* \approx 3.106 \]
   • This means that for a 99.8% confidence level and 11 degrees of freedom, the t-critical value is approximately 3.106.

Summary

The t-critical value is a crucial component in our margin of error calculation, which uses \( t^* \) to scale the sample standard deviation to account for how much variability we might expect in our sample estimates. The confidence interval can then be adjusted using this value to reflect the desired level of confidence accurately.