What is the confidence interval range for the proportion of the target population spending 3 hours or more on social media?
Understand the Problem
The question is asking for the specific range of confidence intervals regarding how much of a target population spends 3 hours or more on social media. To solve it, we need to calculate the confidence interval for the given data concerning social media usage.
Answer
The confidence interval for the target population spending 3 hours or more on social media is $[CI_{lower}, CI_{upper}]$.
Answer for screen readers
The confidence interval for the proportion of the target population spending 3 hours or more on social media is given by the range $[CI_{lower}, CI_{upper}]$.
Steps to Solve
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Collect the Data
Gather the necessary data, which includes the sample size ($n$), the sample proportion ($p$) of the population spending 3 hours or more on social media, and the confidence level (often 95%). -
Calculate the Sample Proportion
If you have raw data, compute the sample proportion using the formula:
$$ p = \frac{x}{n} $$
where $x$ is the number of individuals in the sample who spend 3 hours or more on social media. -
Determine the Z-Score
Select the Z-score corresponding to the desired confidence level. For a 95% confidence level, the Z-score is approximately 1.96. -
Calculate the Standard Error
Calculate the standard error (SE) using the formula:
$$ SE = \sqrt{\frac{p(1-p)}{n}} $$ -
Calculate the Margin of Error
Compute the margin of error (ME) with the formula:
$$ ME = Z \cdot SE $$
where $Z$ is the Z-score you found earlier. -
Determine the Confidence Interval
Finally, calculate the confidence interval with the formulas:
$$ CI_{lower} = p - ME $$
$$ CI_{upper} = p + ME $$
The confidence interval for the proportion of the target population spending 3 hours or more on social media is given by the range $[CI_{lower}, CI_{upper}]$.
More Information
The confidence interval provides a range of values that is likely to contain the true population proportion, giving a better understanding of social media usage habits among the target population.
Tips
- Forgetting to ensure that the sample proportion must be between 0 and 1.
- Miscalculating the standard error, especially if using an incorrect proportion.
- Using the wrong Z-score for different confidence levels.
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