Question

1. The town of Hayward (California) has about 50,000 registered voters. A political scientist takes a simple random sample of 500 of these voters. In the sample, the breakdown by party affiliation is Republican 115 Democrat 331 Independent 54
The range from _______ to __________ is a 95% confidence interval for the percentage of independents among _________________________. Fill in the first two blanks with numbers. Fill in the last blank with one of the following two options: "All the 50,000 voters in the population", "500 voters in the sample".

Answer 1

Answer:

The range from 8.08% to 13.52% is a 95% confidence interval for the percentage of independents among the 50000 registered votersof the town of Hayward

Step-by-step explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$p$ represent the real population proportion of interest

$\hat p =\frac{54}{500}=0.108$ represent the estimated proportion for the sample  of independents

n=500 is the sample size required (variable of interest)

$z$ represent the critical value for the margin of error

The population proportion have the following distribution

$p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})$

The confidence interval would be given by this formula

$\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}$

For the 95% confidence interval the value of $\alpha=1-0.95=0.05$ and $\alpha/2=0.025$, with that value we can find the quantile required for the interval in the normal standard distribution.

$z_{\alpha/2}=1.96$

And replacing into the confidence interval formula we got:

$0.108 - 1.96 \sqrt{\frac{0.108(1-0.108)}{500}}=0.0808$

$0.108 + 1.96 \sqrt{\frac{0.108(1-0.108)}{500}}=0.1352$

And the 95% confidence interval would be given (0.0808;0.1352).

The range from 8.08% to 13.52% is a 95% confidence interval for the percentage of independents among the 50000 registered votersof the town of Hayward