Question

A certain region would like to estimate the proportion of voters who intend to participate in upcoming elections. A pilot sample of 25 voters found that 15 of them intended to vote in the election. Determine the additional number of voters that need to be sampled to construct a 98​% interval with a margin of error equal to 0.06 to estimate the proportion.

The region should sample _______________ additional voters. ​(Round up to the nearest​ integer.)

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A tire manufacturer would like to estimate the average tire life of its new​ all-season light truck tire in terms of how many miles it lasts. Determine the sample size needed to construct a 96​% confidence interval with a margin of error equal to 3,200 miles. Assume the standard deviation for the tire life of this particular brand is 7,000 miles.

The sample size needed is____ . ​(Round up to the nearest​ integer.)

Answer 1

1)
The following information is provided,
Significance Level, α = 0.02, Margin of Error, E = 0.06

The provided estimate of proportion p is, p = 0.6
The critical value for significance level, α = 0.02 is 2.33.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.6*(1 - 0.6)*(2.33/0.06)^2
n = 361.93

Therefore, the sample size needed to satisfy the condition n >= 361.93 and it must be an integer number, we conclude that the minimum required sample size is n = 362
Ans : Sample size, n = 362

Additional number of voters = 362 - 25 = 337

b)

The following information is provided,
Significance Level, α = 0.04, Margin or Error, E = 3200, σ = 7000

The critical value for significance level, α = 0.04 is 2.05.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.05 * 7000/3200)^2
n = 20.11

Therefore, the sample size needed to satisfy the condition n >= 20.11 and it must be an integer number, we conclude that the minimum required sample size is n = 21
Ans : Sample size, n = 21