8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

2:05 minutes

Problem 7.1.37b

Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

b. Assume that 11% of consumers have a smartphone and plan to upgrade to a new model.

Verified step by step guidance

Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is:

n = (Z² * p * (1 - p)) / E²

, where

n

is the sample size,

Z

is the z-score corresponding to the confidence level,

p

is the estimated population proportion, and

E

is the margin of error.

Step 2: Determine the values for the variables in the formula. The confidence level is 90%, so the z-score corresponding to 90% confidence is approximately 1.645. The estimated population proportion

p

is 0.11 (11%), and the margin of error

E

is 0.03 (3%).

Step 3: Substitute the values into the formula. The equation becomes:

n = (1.645² * 0.11 * (1 - 0.11)) / 0.03²

Step 4: Simplify the numerator. Calculate

1.645²

, then multiply it by

0.11

and

(1 - 0.11)

. This will give you the value of the numerator.

Step 5: Simplify the denominator. Calculate

0.03²

. Then divide the numerator by the denominator to find the required sample size

n

. Round up to the nearest whole number, as sample size must be a whole number.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical method used to calculate the number of observations or replicates needed in a survey to achieve a desired level of confidence and precision. In this context, it involves using the estimated proportion of consumers who plan to purchase the new iPhone and the margin of error to determine how many individuals need to be surveyed to ensure that the results are reliable.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the estimated range. A 90% confidence level means that if the survey were repeated multiple times, 90% of the time the calculated confidence interval would contain the true population percentage. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

The margin of error indicates the range within which the true population percentage is expected to fall, based on the survey results. In this case, a margin of error of three percentage points means that the estimated percentage from the survey could be three points higher or lower than the actual percentage. This concept is essential for assessing the precision of the survey findings.

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Textbook Question

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b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

Textbook Question

Airline Seating You are the operations manager for American Airlines and you are considering a higher fare level for passengers in aisle seats. You want to estimate the percentage of passengers who now prefer aisle seats. How many randomly selected air passengers must you survey? Assume that you want to be 95% confident that the sample percentage is within 2.5 percentage points of the true population percentage.

a. Assume that nothing is known about the percentage of passengers who prefer aisle seats.

Textbook Question

Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?

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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

Textbook Question

c. Given that the required sample size is relatively small, could you simply survey the people that you know?

Textbook Question

Women Who Give Birth An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 99% confident that the estimated percentage is in error by no more than two percentage points?

c. What is wrong with surveying randomly selected adult women?

Textbook Question

One-Sided Confidence Interval A one-sided claim about a population proportion is a claim that the proportion is less than (or greater than) some specific value. Such a claim can be formally addressed using a one-sided confidence interval for p, which can be expressed as p<p+E or p>p-E, where the margin of error E is modified by replacing za/2 with za. (Instead of dividing between two tails of the standard normal distribution, put all of it in one tail.) The Chapter Problem refers to a Sallie Mae survey of 950 undergraduate students, and 53% of the survey subjects take online courses. Use that data to construct a one-sided 95% confidence interval that would be suitable for helping to determine whether the percentage of all undergraduates who take online courses is greater than 50%.

Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

a. Assume that nothing is known about the rate of e-cigarette usage among adults.

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