Questions: In an election, suppose that 60% of voters support creating a new fire district. If we poll 139 of these voters at random, the probability distribution for the proportion of the polled voters that support creating a new fire district can be modeled by the normal distribution pictured below. Complete the boxes accurate to two decimal places.

Solution Steps

• Proportion of voters supporting the new fire district (p) = 0.60
• Sample size (n) = 139

The mean (μ) of the sampling distribution of the sample proportion is equal to the population proportion (p). \[ \mu = p = 0.60 \]

The standard deviation (σ) of the sampling distribution of the sample proportion is calculated using the formula: \[ \sigma = \sqrt{\frac{p(1-p)}{n}} \] \[ \sigma = \sqrt{\frac{0.60 \times 0.40}{139}} \] \[ \sigma = \sqrt{\frac{0.24}{139}} \] \[ \sigma = \sqrt{0.001726} \] \[ \sigma \approx 0.04154 \]

The normal distribution is centered at the mean (0.60) and extends three standard deviations on either side. The values for the boxes are:

• Mean (center): 0.60
• One standard deviation below the mean: \( 0.60 - 0.04154 \approx 0.56 \)
• One standard deviation above the mean: \( 0.60 + 0.04154 \approx 0.64 \)
• Two standard deviations below the mean: \( 0.60 - 2 \times 0.04154 \approx 0.52 \)
• Two standard deviations above the mean: \( 0.60 + 2 \times 0.04154 \approx 0.68 \)

Final Answer