A politician is preparing to speak to voters in Connecticut. He DOES NOT favor gun control laws. However, a survey of 1200 voters from this state resulted in 984 voters that supported some level of gun control laws. (Assume a good random sample was taken).
a. Using the survey data given construct a 99% confidence interval for the true proportion of voters in this state that supports some level of gun control laws.
b. You are his political advisor. He asked you if he should bring up his position on gun control laws during his speech to voters in this state. Interpret the confidence interval given in part a. for him in a clear statistical inference. Also, tell how you advise him.
a. To construct a confidence interval for the true proportion of voters in Connecticut who support some level of gun control laws, we can use the formula for confidence interval of a proportion.
The formula is given as:
CI = p̂ ± Z * sqrt((p̂ * (1 - p̂))/n),
Where:
CI = Confidence Interval
p̂ = Sample proportion
Z = Z-score corresponding to the desired confidence level
n = Sample size
Given data:
Sample size (n) = 1200
Number of voters supporting some level of gun control laws (x) = 984
p̂ = x/n = 984/1200 = 0.82 (rounded to two decimal places)
Now, let's find the Z-score using a 99% confidence level. A 99% confidence level corresponds to an alpha value of 0.01. Half of the remaining probability is divided equally on both ends of the distribution, so we have an area of 0.005 at each tail.
Looking up the Z-score from the standard normal distribution table or using a calculator, the Z-score for a 99% confidence level is approximately 2.576.
Substituting the values into the formula:
CI = 0.82 ± 2.576 * sqrt((0.82 * (1 - 0.82))/1200)
Calculating the values:
CI = 0.82 ± 2.576 * sqrt((0.82 * 0.18)/1200)
= 0.82 ± 2.576 * sqrt(0.1476/1200)
= 0.82 ± 2.576 * sqrt(0.000123)
Using a calculator:
CI = 0.82 ± 2.576 * 0.0111
= 0.82 ± 0.0287
Therefore, the 99% confidence interval for the true proportion of voters in Connecticut who support some level of gun control laws is approximately (0.791, 0.849).
b. The confidence interval (0.791, 0.849) means that we are 99% confident that the true proportion of voters in Connecticut who support some level of gun control laws falls within this range (between 0.791 and 0.849).
As a political advisor, the politician should consider several factors in deciding whether to bring up his position on gun control laws during his speech.
1. First, the confidence interval indicates that the proportion of voters supporting some level of gun control laws is estimated to be between 0.791 and 0.849. This suggests that the majority of voters in Connecticut may have a favorable view towards gun control laws.
2. However, it is important to note that the confidence interval is based on a sample and there is a margin of error. The true proportion may still differ slightly from the estimated range.
3. The politician should also consider the political climate, his target audience, and the potential impact of taking a stand against popular opinion. If he believes that his stance aligns with the beliefs and values of a significant portion of Connecticut voters, he may choose to bring up his position on gun control laws during his speech.
4. Alternatively, if he believes that the issue of gun control is contentious and could greatly divide the voters, he may decide to focus on other topics or find a way to address the concerns of both sides, promoting unity and understanding.
Ultimately, the decision should be based on a careful analysis of the political landscape and considering the potential consequences. The confidence interval can provide valuable information, but it should not be the sole determinant in deciding the politician's course of action.
To construct a confidence interval for the true proportion of voters in Connecticut that support some level of gun control laws, we can use the sample proportion and the formula for a confidence interval for a proportion.
a. The formula for a confidence interval for a proportion is:
CI = p̂ ± z * √((p̂ * (1 - p̂))/n)
where:
- p̂ is the sample proportion (984/1200 = 0.82)
- z is the critical z-value for the desired confidence level (99%, two-tailed test has a z = 2.58 for 99% confidence level)
- n is the sample size (1200)
Plugging in the values:
CI = 0.82 ± 2.58 * √((0.82 * (1 - 0.82))/1200)
= 0.82 ± 2.58 * √((0.82 * 0.18)/1200)
= 0.82 ± 2.58 * √(0.1476/1200)
= 0.82 ± 2.58 * √(0.000123)
Calculating the square root of 0.000123, we get:
CI = 0.82 ± 2.58 * 0.01109
= 0.82 ± 0.0286
Therefore, the 99% confidence interval for the true proportion of voters in Connecticut who support some level of gun control laws is approximately (0.7914, 0.8506).
b. The confidence interval obtained in part a represents the range of values within which we can be 99% confident that the true proportion of voters in Connecticut who support some level of gun control laws lies. Specifically, we are 99% confident that the proportion falls between 0.7914 and 0.8506.
As a political advisor, the decision of whether the politician should bring up his position on gun control laws during his speech depends on how comfortable he is with the potential range of voter support. If he believes that a majority of voters fall within the 99% confidence interval (0.7914 - 0.8506), then it may be a good idea for him to discuss his position during the speech. However, if he feels that the proportion is closer to the lower or upper limit of the interval, he may want to consider the potential impact on his voter base and adjust his approach accordingly. Ultimately, the decision should be based on his own judgment, taking into account the statistical information provided.