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A sample of voters in a certain city was asked whether they planned to vote for the incumbent mayor in the next election. The following display from a TI-84 Plus calculator presents a 99% confidence interval for the population proportion that plan to vote for the incumbent mayor.

1-PropZInt
(.34795, .66743)
^p=.5076923077
n = 65
a) Fill in the blanks. We are ______% confident that the population proportion is between ______ and ______. Round the answers to three decimal places as needed.

3 answers

To fill in the blanks based on the provided confidence interval display, we can extract the necessary information from the interval and context:

  • The confidence level is given as 99%, so we are 99% confident.
  • The lower bound of the confidence interval is 0.34795, which rounds to 0.348 to three decimal places.
  • The upper bound of the confidence interval is 0.66743, which rounds to 0.667 to three decimal places.

Therefore, the filled-in statement will read:

We are 99% confident that the population proportion is between 0.348 and 0.667.

(b) Use the information in the display to construct a 95% confidence interval for p. Round the answers to three decimal places.

To construct a 95% confidence interval for the population proportion \( p \), we will use the sample proportion and the standard error of the sample proportion. We know the following from the given information:

  • Sample proportion \( \hat{p} = 0.5077 \) (as provided in the display).
  • Sample size \( n = 65 \).

Now, to calculate the 95% confidence interval, we will take the following steps:

  1. Calculate the standard error (SE) of the sample proportion: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5076923077 \times (1 - 0.5076923077)}{65}}. \]

  2. Calculate the critical value (z) for 95% confidence*: The critical z-value for a 95% confidence level is approximately 1.96.

  3. Use the formula for the confidence interval: \[ CI = \hat{p} \pm z^* \cdot SE. \]

Now, let's go through the calculations:

  1. Calculate SE: \[ SE = \sqrt{\frac{0.5077 \times (1 - 0.5077)}{65}} \approx \sqrt{\frac{0.5077 \times 0.4923}{65}} \approx \sqrt{\frac{0.249173911}{65}} \approx \sqrt{0.003833} \approx 0.0619. \]

  2. Calculate the confidence interval: \[ CI = 0.5077 \pm 1.96 \cdot 0.0619. \]

    Calculate the margin of error: \[ Margin of Error = 1.96 \times 0.0619 \approx 0.1211. \]

    Now calculate the lower and upper bounds of the interval: \[ Lower Bound = 0.5077 - 0.1211 \approx 0.3866, \] \[ Upper Bound = 0.5077 + 0.1211 \approx 0.6288. \]

  3. Round to three decimal places:

    • Lower Bound: 0.387
    • Upper Bound: 0.629

Thus, the 95% confidence interval for \( p \) is: (0.387, 0.629).