What would the new confidence interval be if the average reading time is 25 minutes with a standard deviation of 20 minutes for a 99% confidence level?
Understand the Problem
The question is asking how to calculate the new confidence interval for a given average reading time and standard deviation at a specified confidence level. Specifically, it seeks to find the range of values within which the true average is likely to fall.
Answer
The confidence interval is $ \left(\bar{x} - ME, \bar{x} + ME\right) $.
Answer for screen readers
The confidence interval is given by the range $$ \left(\bar{x} - ME, \bar{x} + ME\right) $$
Steps to Solve
- Identify Given Values First, note the values provided in the problem:
- Sample mean ($\bar{x}$)
- Standard deviation ($s$)
- Sample size ($n$)
- Desired confidence level (usually expressed as a percentage like 95% or 99%)
- Determine the Z-Score or T-Score Based on the sample size and confidence level, you'll need to look up the Z-score or T-score in a statistical table:
- For larger samples (typically $n > 30$), use the Z-score.
- For smaller samples (typically $n \leq 30$), use the T-score.
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Calculate the Standard Error Calculate the Standard Error (SE) using the formula: $$ SE = \frac{s}{\sqrt{n}} $$ Where $s$ is the sample standard deviation and $n$ is the sample size.
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Calculate the Margin of Error Multiply the standard error by the Z-score or T-score to find the Margin of Error (ME): $$ ME = Z \cdot SE \quad \text{or} \quad ME = T \cdot SE $$ Depending on whether you used the Z-score or T-score.
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Find the Confidence Interval Finally, use the margin of error to find the confidence interval: $$ \text{Lower Limit} = \bar{x} - ME $$ $$ \text{Upper Limit} = \bar{x} + ME $$
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State the Result Present the final confidence interval value, indicating the range where the true average likely falls.
The confidence interval is given by the range $$ \left(\bar{x} - ME, \bar{x} + ME\right) $$
More Information
Confidence intervals provide a range of values that estimate where the true population parameter lies. The wider the interval, the less confidence you have that the estimate is close to the true average. The confidence level indicates how often this interval will capture the true parameter if you were to repeat the process multiple times.
Tips
- Forgetting to use the correct score (Z or T) based on the sample size.
- Miscalculating the standard error or margin of error.
- Failing to interpret the results in the context of the problem.
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