What is the minimum number of consumers Timex should survey …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

Select the best answer. Many television viewers express doubts about the validity of certain commercials. In an attempt to answer their critics, Timex Group USA wishes to estimate the proportion of consumers who believe what is shown in Timex television commercials. Let p represent the true proportion of consumers who believe what is shown in Timex television commercials. What is the smallest number of consumers that Timex can survey to guarantee a margin error of 0.05 or less at a 99% confidence level? (a) 550 (b) 600 (c) 650 (d) 700 (e) 750

Short Answer

To determine the sample size for a 99% confidence level with a 5% margin of error, first, identify the necessary parameters: Z critical value of 2.58 and assume a proportion of 0.5. Using the sample size formula, the calculated sample size is 665.64, rounded to 700 consumers for accurate survey results.

Step-by-Step Solution

Step 1: Understand the Confidence Level and Margin of Error

Before calculating the sample size, it’s crucial to know the desired confidence level and margin of error (ME). In this case, we’re looking for a 99% confidence level with a margin of error of 0.05. This means that the results of the survey should be accurate within 5% of the true population value at a confidence level of 99%, ensuring high reliability in the data collected.

Step 2: Use the Sample Size Formula

To find the sample size, apply the formula: n = (Z critical / ME)² * p(1 – p). The Z critical value for a 99% confidence level is 2.58. Since we don’t have a proportion provided, we assume it as 50% (0.5) for maximum variability. This is essential for determining the largest sample size needed, thereby ensuring that our estimate is robust.

  • Set Z critical = 2.58
  • Set ME = 0.05
  • Assume p = 0.5

Step 3: Calculate and Round the Sample Size

Now, plug the values into the formula. You calculate as follows: (2.58 / 0.05)² * 0.5(1 – 0.5) = 665.64. After calculating, round this value to the nearest hundred for practical purposes. Hence, the smallest number of consumers that Timex can survey is 700 to meet the criteria of the margin of error and confidence level.

Related Concepts

Confidence Level

The degree of certainty that the true population parameter lies within the confidence interval, typically expressed as a percentage (e.g., 99%)

Margin Of Error (Me)

The maximum amount by which the sample estimate is expected to differ from the true population value, indicating the precision of the estimate, usually expressed as a percentage (e.g., 5%)

Z Critical Value

The value that corresponds to a specific confidence level in a standard normal distribution, used in sample size calculations to determine how many standard deviations away the estimate is from the mean, critical for applying the sample size formula.

Table Of Contents
  1. Select the best answer. Many television viewers express doubts about the validity of certain commercials. In an attempt to answer their critics, Timex Group USA wishes to estimate the proportion of consumers who believe what is shown in Timex television commercials. Let p represent the true proportion of consumers who believe what is shown in Timex television commercials. What is the smallest number of consumers that Timex can survey to guarantee a margin error of 0.05 or less at a 99% confidence level? (a) 550 (b) 600 (c) 650 (d) 700 (e) 750

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