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A simple random sample of eight college freshmen were asked how many hours of sleep they typically got per night. The results were

8.5
24
9
7.5
8
6.5
8.5
6

(a) The data contain an outlier that is clearly a mistake. Eliminate the outlier, then construct a 98% confidence interval for the mean amount of sleep from the remaining values. Round the answers to at least two decimal places.

3 answers

To determine the outlier and construct a 98% confidence interval for the mean amount of sleep, we first eliminate the outlier from the dataset. The given data is:

  • 8.5
  • 24 (this appears to be an outlier)
  • 9
  • 7.5
  • 8
  • 6.5
  • 8.5
  • 6

After removing the outlier (24), the remaining data is:

  • 8.5
  • 9
  • 7.5
  • 8
  • 6.5
  • 8.5
  • 6

Now we'll calculate the mean and standard deviation of the remaining values:

  1. Calculate the Mean (\(\bar{x}\)): \[ \bar{x} = \frac{\sum x}{n} \]

    Where \( n \) is the number of values, and \( x \) is each individual value.

    \[ \bar{x} = \frac{8.5 + 9 + 7.5 + 8 + 6.5 + 8.5 + 6}{7} = \frac{54.0}{7} \approx 7.7143 \]

  2. Calculate the Standard Deviation (s): \[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]

    First, we'll calculate each squared difference:

    • \( (8.5 - 7.7143)^2 \approx 0.6102 \)
    • \( (9 - 7.7143)^2 \approx 1.6494 \)
    • \( (7.5 - 7.7143)^2 \approx 0.0457 \)
    • \( (8 - 7.7143)^2 \approx 0.0802 \)
    • \( (6.5 - 7.7143)^2 \approx 1.4784 \)
    • \( (8.5 - 7.7143)^2 \approx 0.6102 \)
    • \( (6 - 7.7143)^2 \approx 2.9347 \)

    Now we sum these squared differences: \[ \sum (x_i - \bar{x})^2 \approx 0.6102 + 1.6494 + 0.0457 + 0.0802 + 1.4784 + 0.6102 + 2.9347 \approx 7.5088 \]

    Plugging this into the standard deviation formula: \[ s = \sqrt{\frac{7.5088}{6}} \approx \sqrt{1.2515} \approx 1.1187 \]

  3. Calculate the 98% Confidence Interval:

    The formula for the confidence interval is: \[ \bar{x} \pm t^* \times \frac{s}{\sqrt{n}} \]

    We need to find the t-value for a 98% confidence interval with \( n-1 = 6 \) degrees of freedom. From the t-table, the critical t-value (\( t^* \)) for 98% confidence and 6 degrees of freedom is approximately 2.447.

    Now, we calculate the margin of error: \[ \text{Margin of Error} = t^* \times \frac{s}{\sqrt{n}} = 2.447 \times \frac{1.1187}{\sqrt{7}} \approx 2.447 \times 0.4226 \approx 1.0353 \]

    Finally, the confidence interval is: \[ \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right) = \left( 7.7143 - 1.0353, 7.7143 + 1.0353 \right) \approx (6.68, 8.75) \]

Thus, the 98% confidence interval for the mean amount of sleep is approximately:

\[ \boxed{(6.68, 8.75)} \]

can you explain more on how to get the t-value

Certainly! The t-value is determined using the Student's t-distribution, which is commonly used when estimating the mean of a normally distributed population when the sample size is small (typically \( n < 30 \)) and the population standard deviation is unknown.

How to Find the t-value

  1. Determine the Confidence Level: In this case, you are looking for a 98% confidence level. This means that we want to find the critical value that leaves 2% in the tails of the distribution (1% in each tail).

  2. Determine the Degrees of Freedom (df): The degrees of freedom for a single sample is calculated as: \[ df = n - 1 \] where \( n \) is the sample size. In your case, \( n = 7 \), so: \[ df = 7 - 1 = 6 \]

  3. Consult a t-Distribution Table or Use a Statistical Software: Once you have the confidence level and the degrees of freedom, you can look up the corresponding t-value in a t-table.

    • Use a t-table: In the t-table, find the row corresponding to \( df = 6 \). Then look for the column that corresponds to the desired confidence level. Since you want a two-tailed test for a 98% confidence interval, you want \( \alpha = 0.02 \) (which means 1% in each tail).

    • The t-table will usually indicate values for common confidence levels (e.g., 90%, 95%, 99%). Look for the value that corresponds to a probability of \( 0.01 \) in one tail (or \( 0.02 \) total), which results in the critical t-value.

    • For a confidence level of 98% and \( df = 6 \), you typically find \( t^* \approx 2.447 \). This value is rounded and may vary slightly depending on the source of your t-table.

  4. Using Statistical Software or Online Calculators: Alternatively, you can use software such as R, Python, Excel, or even online calculators to find the t-value:

    • In R, you can use the command qt(0.01, df=6) to get the t-value.
    • In Python, using the scipy.stats library, you can use scipy.stats.t.ppf(0.99, 6) to find the same value.

Summary of the Steps to Obtain the t-value

  • Set your confidence level (e.g., 98%).
  • Calculate degrees of freedom (\( df = n - 1 \)).
  • Look it up in a t-table or use software to find the t-value for \( df = 6 \) under the specified confidence level.

Following this process gives you the critical t-value needed to construct confidence intervals for your sample mean. In your particular case, that t-value was approximately 2.447 for a 98% confidence interval with 6 degrees of freedom.