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To determine how well a new method of teaching vocabulary is working in a certain elementary school, education researchers plan to give a vocabulary test to a sample of 40 sixth graders. Assume the sample standard deviation is s=8. The researchers plan to compute the sample mean x, then construct a 96% confidence interval for the population mean test score. Interpret the interval.

use a t-84 calculator

3 answers

To construct a 96% confidence interval for the population mean test score based on your sample, we can use the following formula for the confidence interval:

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

Where:

  • \( \bar{x} \) is the sample mean.
  • \( t^* \) is the t-score corresponding to the desired confidence level and degrees of freedom.
  • \( s \) is the sample standard deviation.
  • \( n \) is the sample size.

Steps to Calculate the Confidence Interval:

  1. Sample Size (n): 40

  2. Sample Standard Deviation (s): 8

  3. Degrees of Freedom (df): \( n - 1 = 40 - 1 = 39 \)

  4. Determine \( t^ \)*:

    • For a 96% confidence level and 39 degrees of freedom, you can find \( t^* \). Using a t-table or a calculator like the TI-84, the critical t-value can typically be found using the invT function.

    For a 96% confidence interval with 4% in the tails combined (2% in each tail), you would calculate: \[ t^* = \text{invT}(0.02, 39) \]

    Using the TI-84, input invT(0.02, 39) to find \( t^* \).

  5. Calculating Standard Error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{40}} \approx \frac{8}{6.32} \approx 1.27 \]

  6. Plug in the values: Now, calculate the confidence interval using the formula mentioned above once you have \( \bar{x} \) and \( t^* \).

  7. Final Calculation: The confidence interval will be in the form: \[ (\bar{x} - t^* \cdot SE, \bar{x} + t^* \cdot SE) \]

Interpretation of the Confidence Interval:

Once you calculate the confidence interval using your sample mean \( \bar{x} \), you would interpret it in the context of the study. Suppose your calculated 96% confidence interval was \( (C_1, C_2) \), you would say:

"We are 96% confident that the true population mean test score for sixth graders taught with the new vocabulary teaching method is between \( C_1 \) and \( C_2 \)."

Note: Please ensure you obtain the sample mean from your testing with the 40 students to complete the calculation.

Let m be the margin of error for a 96% confidence interval. We conclude that [blank] % of all samples will produce a confidence interval x-m < [blank] < x+m that covers the true [blank] mean.

To fill in the blanks in your statement regarding margins of error and confidence intervals, we can structure the conclusions as follows:

Let \( m \) be the margin of error for a 96% confidence interval. We conclude that 96% of all samples will produce a confidence interval \( \bar{x} - m < \mu < \bar{x} + m \) that covers the true population mean.

Breakdown of the Statement:

  • 96%: This is the confidence level, indicating the proportion of confidence intervals that will capture the true mean if you were to draw many samples from the population.
  • \(\mu\): This symbolizes the true population mean you are estimating.
  • The interval notation \( \bar{x} - m < \mu < \bar{x} + m \) describes how the confidence interval is constructed around the sample mean \( \bar{x} \).

Therefore, filling in the blanks:

"We conclude that 96% of all samples will produce a confidence interval \( \bar{x} - m < \mu < \bar{x} + m \) that covers the true population mean."