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To determine the correct interpretation of the given [tex]$95 \%$[/tex] confidence interval for the mean difference in height of pairs of identical twins, we need to understand what a confidence interval represents.
A [tex]$95 \%$[/tex] confidence interval provides a range of values within which we can be [tex]$95 \%$[/tex] confident that the true population parameter lies. In this case, the parameter of interest is the true mean difference in height between Twin 1 and Twin 2.
Given the data, we need to interpret the confidence interval [tex]\((-0.823, 0.573)\)[/tex]. Here's a step-by-step solution:
1. Identify the Population Parameter: We are interested in the mean difference in height between pairs of identical twins. Note that this is a population parameter.
2. Sample and Confidence Interval: The sample consists of heights from 8 pairs of twins. The calculated [tex]$95 \%$[/tex] confidence interval for the mean difference (Twin 1 - Twin 2) is [tex]\((-0.823, 0.573)\)[/tex].
3. Interpretation of the Confidence Interval:
- The confidence interval captures the range within which the true mean difference in height between Twin 1 and Twin 2 is likely to fall.
- Specifically, with [tex]$95 \%$[/tex] confidence, we believe the true mean difference is between [tex]\(-0.823\)[/tex] inches and [tex]\(0.573\)[/tex] inches.
4. Finding the Correct Choice:
- The interval needs to capture the true mean difference, not just the mean heights.
- The interval should pertain to the population parameter, not the sample statistics.
Reviewing the options:
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins." - This option is incorrect because it speaks about the height of twins, not the difference in their heights.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins in this sample." - This option is incorrect because it incorrectly describes the interval as capturing the sample mean, not the population parameter.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins." - This option is correct because it correctly states that the interval captures the true mean difference in height.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of the twins in this sample." - This option is incorrect because it refers to the sample, not the population.
Therefore, the correct interpretation of the [tex]$95 \%$[/tex] confidence interval is: "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins."
A [tex]$95 \%$[/tex] confidence interval provides a range of values within which we can be [tex]$95 \%$[/tex] confident that the true population parameter lies. In this case, the parameter of interest is the true mean difference in height between Twin 1 and Twin 2.
Given the data, we need to interpret the confidence interval [tex]\((-0.823, 0.573)\)[/tex]. Here's a step-by-step solution:
1. Identify the Population Parameter: We are interested in the mean difference in height between pairs of identical twins. Note that this is a population parameter.
2. Sample and Confidence Interval: The sample consists of heights from 8 pairs of twins. The calculated [tex]$95 \%$[/tex] confidence interval for the mean difference (Twin 1 - Twin 2) is [tex]\((-0.823, 0.573)\)[/tex].
3. Interpretation of the Confidence Interval:
- The confidence interval captures the range within which the true mean difference in height between Twin 1 and Twin 2 is likely to fall.
- Specifically, with [tex]$95 \%$[/tex] confidence, we believe the true mean difference is between [tex]\(-0.823\)[/tex] inches and [tex]\(0.573\)[/tex] inches.
4. Finding the Correct Choice:
- The interval needs to capture the true mean difference, not just the mean heights.
- The interval should pertain to the population parameter, not the sample statistics.
Reviewing the options:
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins." - This option is incorrect because it speaks about the height of twins, not the difference in their heights.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins in this sample." - This option is incorrect because it incorrectly describes the interval as capturing the sample mean, not the population parameter.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins." - This option is correct because it correctly states that the interval captures the true mean difference in height.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of the twins in this sample." - This option is incorrect because it refers to the sample, not the population.
Therefore, the correct interpretation of the [tex]$95 \%$[/tex] confidence interval is: "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins."
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