1. The town of Hayward (California) has about 50,000 registered voters. A political scientist takes a simple random sample of 500 of these v

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”.

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Lydia 1 week 2021-09-14T21:17:12+00:00 1 Answer 0

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    2021-09-14T21:19:08+00:00

    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

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