Answer :
The probability that the proportion of polled voters supporting the incumbent candidate is less than 0.37 is approximately 0.6915, and the probability that the proportion is between 0.32 and 0.38 is approximately 0.3004.
To complete the boxes accurately, we need to use the given information and the normal distribution to determine the appropriate values.
The normal distribution is characterized by its mean (μ) and standard deviation (σ). In this case, the mean proportion of voters supporting the incumbent candidate is given as 35% or 0.35. However, we need to calculate the standard deviation.
To calculate the standard deviation, we can use the formula:
σ = √(p(1-p)/n),
where p is the proportion supporting the incumbent candidate (0.35) and n is the sample size (140).
σ = √(0.35(1-0.35)/140)
= √(0.35(0.65)/140)
= √(0.2275/140)
≈ √0.001625
≈ 0.04031.
Now that we have the standard deviation, we can determine the probabilities in the given normal distribution.
In the first box, we need to find the probability that the proportion is less than 0.37. We can use the standard normal distribution table or a calculator to find the corresponding z-score and its probability. The z-score can be calculated using the formula:
z = (x - μ)/σ,
where x is the value (0.37), μ is the mean (0.35), and σ is the standard deviation (0.04031).
z = (0.37 - 0.35)/0.04031
≈ 0.02/0.04031
≈ 0.4953.
Using the z-table or calculator, we can find that the probability corresponding to a z-score of 0.4953 is approximately 0.6915.
Therefore, the probability that the proportion of polled voters supporting the incumbent candidate is less than 0.37 is approximately 0.6915.
In the second box, we need to find the probability that the proportion is between 0.32 and 0.38. We can repeat the process for each value and find the corresponding z-scores:
z1 = (0.32 - 0.35)/0.04031 ≈ -0.07448,
z2 = (0.38 - 0.35)/0.04031 ≈ 0.7445.
Using the z-table or calculator, we can find the probabilities corresponding to these z-scores:
P(z < -0.07448) ≈ 0.4698,
P(z < 0.7445) ≈ 0.7702.
The probability that the proportion of polled voters supporting the incumbent candidate is between 0.32 and 0.38 is approximately 0.7702 - 0.4698 ≈ 0.3004.
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