91Ó°ÊÓ

When conducting a survey or a study, the sample proportion is a crucial statistic that helps us understand the makeup of our sample. In this context, the sample proportion is used to estimate the true proportion of a larger population. It's basically the proportion of individuals in a sample with a certain characteristic, which we use to represent the whole.

In the given exercise, two specific examples are provided: the proportion who find it hard to give up their computer, and the proportion who would find it hard to give up their television. These proportions are respectively 0.74 and 0.48. To arrive at these proportions, we divide the number of respondents with the characteristic by the total sample size.

Using the responses, out of 1677 participants, 74% find it hard to give up their computer. We can express this as:

• Sample Proportion = 0.74
• Sample Size = 1677

Understanding sample proportion is fundamental because it provides the basis for further statistical work, such as computing confidence intervals.

The standard error of a sample proportion is an important part of understanding the variability or dispersion of the sample data around the true population proportion. It tells us how much the sample proportion can be expected to fluctuate from the actual population proportion.

To calculate standard error (SE), we use the formula:\[SE = \sqrt{\frac{p(1-p)}{n}}\], where:

• \(p\)is the sample proportion.
• \(n\)is the sample size.

For example, in the exercise for part (a), the standard error computation for those who can't give up their computers (\(p = 0.74\)) is:\[SE = \sqrt{\frac{0.74 \times (1-0.74)}{1677}}\]From this, we can grasp the spread of our data, which aids in constructing confidence intervals. Smaller standard error indicates the sample proportion is a more accurate reflection of the true population proportion.

The margin of error is the range of values below and above the sample proportion that represents uncertainty. It accounts for any error due to sampling variability, allowing us to express that we are a certain percentage confident the true proportion lies within this range.

To calculate the margin of error, we multiply the Z-score corresponding to our desired confidence level by the standard error (SE) of our sample proportion. For instance, in a 95% confidence interval, we use a Z-score of 1.96, and for a 99% confidence interval, a Z-score of 2.576.

In the exercise:

• Part (a): Margin of Error = 1.96 \(\times\) SE
• Part (b): Margin of Error = 2.576 \(\times\) SE

This means that in part (b), increasing the confidence level requires a larger margin of error. This is because higher confidence levels widen the interval to ensure the true proportion is captured, acknowledging more uncertainty.