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Answer :
To evaluate which statements are true regarding confidence intervals, we need to understand some fundamental concepts about how confidence intervals work.
The 90% confidence interval for Sample 13 does not indicate that 90% of the Sample 13 data values are between 98.5 and 103.7.
- This statement is true. A 90% confidence interval refers to our confidence that the population parameter (like the mean) lies within this interval. It does not imply that 90% of individual sample data points fall within this range.
The 75% confidence interval for Sample 13 is narrower than the 90% confidence interval for Sample 13.
- This statement is true. Generally, the lower the confidence level, the narrower the interval. A 75% confidence interval is likely to be narrower than a 90% confidence interval because there is less certainty required in the estimate.
This is coincidence; when constructing a confidence interval for a sample, there is no relationship between the level of confidence and the width of the interval.
- This statement is false. There is indeed a relationship: higher confidence levels result in wider intervals, as they require more margin to ensure the parameter is captured within the interval.
From the 75% confidence interval for Sample 13, we know that there is a 75% probability that the population mean is between 99.3 and 102.9.
- This statement is false. Confidence intervals do not express the probability that the population mean lies within the interval. Instead, they mean that if we were to take many samples and create a confidence interval from each sample, we expect about 75% of those intervals to contain the population mean.
If there were a Sample 21 of size n=20 taken from the same population as Sample 13, then the 90% confidence interval for Sample 21 would be wider than the 90% confidence interval for Sample 13.
- This statement is false if Sample 13 has more than 20 data points. Confidence intervals generally become narrower as the sample size increases, assuming the same confidence level and population variance.
None of the choices above are true.
- This option is false, as statements 1 and 2 are true.
In summary, the true statements are:
- The 90% confidence interval for Sample 13 does not indicate that 90% of the Sample 13 data values are between 98.5 and 103.7.
- The 75% confidence interval for Sample 13 is narrower than the 90% confidence interval for Sample 13.
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