Question

A research group conducted an extensive survey of 2975 wage and salaried workers on issues ranging from relationships with their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally representative sample. In response to the question, "What does success mean to you?" 1608 responded, "Personal satisfaction from doing a good job." Let p be the population proportion of all wage and salaried workers who would respond the same way to the stated question. Find a 90% confidence interval for p. (Round your answers to three decimal places.)lower limit

upper limit

Answer 1

90% Confidence interval for population proportion = (0.526, 0.556)

Lower limit of the 90% confidence interval for population proportion of all wage and salaried workers who would respond the same way to the stated question = 0.526

Upper limit of the 90% confidence interval for population proportion of all wage and salaried workers who would respond the same way to the stated question = 0.556

Explanation:

- Extensive survey is carried out on 2975 wage and salaried workers on issues ranging from relationships with their bosses to household chores.

- 1608 responded, "Personal satisfaction from doing a good job."

p is the population proportion of all wage and salaried workers who would respond the same way to the stated question. Find a 90% confidence interval for p.

Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample proportion) ± (Margin of error)

Sample proportion = (1608) ÷ (2975) = 0.5405

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error)

Critical value at 90% confidence interval for sample size of 2975 is obtained from the z-tables. (Large enough sample size)

Critical value = 1.645

standard Error will be calculated thus

Standard error = σₓ = √[p(1-p)/n]

p = 0.5405

n = sample size = 2975

σₓ = √[0.5405×0.4595/2975] = 0.0091368632

σₓ = 0.009137

90% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]

CI = 0.5405 ± (1.645 × 0.009137)

CI = 0.5405 ± 0.015030365

CI = 0.5405 ± 0.0150

90% CI = (0.5255, 0.5555)

90% Confidence interval = (0.526, 0.556)

Hope this Helps!!!