Confidence Interval for a Population Mean - AP Statistics
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Find $t^*$ for $df = 10$ at 95% confidence level.
Find $t^*$ for $df = 10$ at 95% confidence level.
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Approximately 2.228. From $t$-table with $df = 10$ and 95% confidence level.
Approximately 2.228. From $t$-table with $df = 10$ and 95% confidence level.
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Identify the effect of decreasing variability on the confidence interval.
Identify the effect of decreasing variability on the confidence interval.
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It narrows the interval. Less variability reduces standard error and margin of error.
It narrows the interval. Less variability reduces standard error and margin of error.
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Calculate the standard error: $s = 3$, $n = 36$.
Calculate the standard error: $s = 3$, $n = 36$.
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$\frac{3}{\sqrt{36}} = 0.5$. Standard error formula: $s$ divided by square root of $n$.
$\frac{3}{\sqrt{36}} = 0.5$. Standard error formula: $s$ divided by square root of $n$.
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Calculate the interval: $\bar{x} = 50$, $t^* = 2.5$, $s = 10$, $n = 16$.
Calculate the interval: $\bar{x} = 50$, $t^* = 2.5$, $s = 10$, $n = 16$.
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$50 \pm 6.25$. Margin of error: $2.5 \times \frac{10}{\sqrt{16}} = 2.5 \times 2.5 = 6.25$.
$50 \pm 6.25$. Margin of error: $2.5 \times \frac{10}{\sqrt{16}} = 2.5 \times 2.5 = 6.25$.
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How does sample size relate to the standard error?
How does sample size relate to the standard error?
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Larger sample size decreases standard error. Standard error decreases as $n$ increases: $\frac{s}{\sqrt{n}}$.
Larger sample size decreases standard error. Standard error decreases as $n$ increases: $\frac{s}{\sqrt{n}}$.
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Calculate confidence interval: $\bar{x} = 20$, $t^* = 1.96$, $s = 4$, $n = 100$.
Calculate confidence interval: $\bar{x} = 20$, $t^* = 1.96$, $s = 4$, $n = 100$.
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$20 \pm 0.784$. Margin of error: $1.96 \times \frac{4}{\sqrt{100}} = 1.96 \times 0.4 = 0.784$.
$20 \pm 0.784$. Margin of error: $1.96 \times \frac{4}{\sqrt{100}} = 1.96 \times 0.4 = 0.784$.
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Find the degrees of freedom for $n = 15$.
Find the degrees of freedom for $n = 15$.
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- Always $n - 1$ for single-sample $t$-procedures.
- Always $n - 1$ for single-sample $t$-procedures.
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State the formula for a confidence interval for a mean.
State the formula for a confidence interval for a mean.
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$\bar{x} , \pm , t^* \left(\frac{s}{\sqrt{n}}\right)$. Sample mean plus/minus margin of error using $t$-distribution.
$\bar{x} , \pm , t^* \left(\frac{s}{\sqrt{n}}\right)$. Sample mean plus/minus margin of error using $t$-distribution.
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What does $\bar{x}$ represent in the confidence interval formula?
What does $\bar{x}$ represent in the confidence interval formula?
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Sample mean. The average of all observations in the sample.
Sample mean. The average of all observations in the sample.
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What does $t^*$ represent in the confidence interval formula?
What does $t^*$ represent in the confidence interval formula?
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Critical value from the $t$-distribution. Based on confidence level and degrees of freedom.
Critical value from the $t$-distribution. Based on confidence level and degrees of freedom.
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What does $s$ represent in the confidence interval formula?
What does $s$ represent in the confidence interval formula?
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Sample standard deviation. Measures spread of sample observations around sample mean.
Sample standard deviation. Measures spread of sample observations around sample mean.
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What does $n$ represent in the confidence interval formula?
What does $n$ represent in the confidence interval formula?
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Sample size. Total number of observations in the sample.
Sample size. Total number of observations in the sample.
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Identify the condition for using a $t$-distribution in confidence intervals.
Identify the condition for using a $t$-distribution in confidence intervals.
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Population standard deviation is unknown and sample is small. Use when $\sigma$ is unknown, especially with small samples.
Population standard deviation is unknown and sample is small. Use when $\sigma$ is unknown, especially with small samples.
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What is the central limit theorem in context of confidence intervals?
What is the central limit theorem in context of confidence intervals?
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The sampling distribution of the sample mean is approximately normal. Allows use of normal approximation for sample means.
The sampling distribution of the sample mean is approximately normal. Allows use of normal approximation for sample means.
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State the formula for standard error of the mean.
State the formula for standard error of the mean.
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$\frac{s}{\sqrt{n}}$. Standard deviation of the sampling distribution of $\bar{x}$.
$\frac{s}{\sqrt{n}}$. Standard deviation of the sampling distribution of $\bar{x}$.
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What does a 95% confidence level imply?
What does a 95% confidence level imply?
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95% of such intervals will capture the true population mean. Long-run frequency of intervals containing true parameter.
95% of such intervals will capture the true population mean. Long-run frequency of intervals containing true parameter.
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How do you interpret a confidence interval?
How do you interpret a confidence interval?
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A range where the population mean is likely to be found. Plausible values for the unknown population mean.
A range where the population mean is likely to be found. Plausible values for the unknown population mean.
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Identify the effect of increasing sample size on the confidence interval.
Identify the effect of increasing sample size on the confidence interval.
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It narrows the confidence interval. Larger $n$ reduces standard error and margin of error.
It narrows the confidence interval. Larger $n$ reduces standard error and margin of error.
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Identify the conditions for normality in constructing confidence intervals.
Identify the conditions for normality in constructing confidence intervals.
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Sample size large or population distribution approximately normal. Ensures sampling distribution of $\bar{x}$ is approximately normal.
Sample size large or population distribution approximately normal. Ensures sampling distribution of $\bar{x}$ is approximately normal.
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What is the effect of confidence level on interval width?
What is the effect of confidence level on interval width?
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Higher confidence level widens the interval. More confidence requires wider interval to capture parameter.
Higher confidence level widens the interval. More confidence requires wider interval to capture parameter.
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Which term describes the range of values in a confidence interval?
Which term describes the range of values in a confidence interval?
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Margin of error. Half-width of confidence interval around point estimate.
Margin of error. Half-width of confidence interval around point estimate.
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Find the margin of error given $t^* = 2$, $s = 4$, $n = 16$.
Find the margin of error given $t^* = 2$, $s = 4$, $n = 16$.
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$2 \times \frac{4}{\sqrt{16}} = 2$. Margin of error equals $t^* \times$ standard error.
$2 \times \frac{4}{\sqrt{16}} = 2$. Margin of error equals $t^* \times$ standard error.
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How does variability affect the confidence interval?
How does variability affect the confidence interval?
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Greater variability widens the interval. Higher $s$ increases standard error and interval width.
Greater variability widens the interval. Higher $s$ increases standard error and interval width.
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What happens to the interval if sample standard deviation decreases?
What happens to the interval if sample standard deviation decreases?
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The interval becomes narrower. Smaller $s$ reduces standard error and margin of error.
The interval becomes narrower. Smaller $s$ reduces standard error and margin of error.
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Identify the sample mean in the data set: 5, 6, 7, 8, 9.
Identify the sample mean in the data set: 5, 6, 7, 8, 9.
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$\bar{x} = 7$. Sum of values divided by count: $(5+6+7+8+9)/5 = 7$.
$\bar{x} = 7$. Sum of values divided by count: $(5+6+7+8+9)/5 = 7$.
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Which table do you use to find $t^*$ values?
Which table do you use to find $t^*$ values?
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The $t$-distribution table. Critical values depend on confidence level and degrees of freedom.
The $t$-distribution table. Critical values depend on confidence level and degrees of freedom.
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Identify the degrees of freedom for a sample size of 25.
Identify the degrees of freedom for a sample size of 25.
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- Degrees of freedom equals $n - 1 = 25 - 1 = 24$.
- Degrees of freedom equals $n - 1 = 25 - 1 = 24$.
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What is the primary purpose of a confidence interval?
What is the primary purpose of a confidence interval?
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To estimate a population parameter. Uses sample data to infer about population parameters.
To estimate a population parameter. Uses sample data to infer about population parameters.
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What does a wider confidence interval suggest about precision?
What does a wider confidence interval suggest about precision?
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Less precision. Wider intervals indicate greater uncertainty about parameter.
Less precision. Wider intervals indicate greater uncertainty about parameter.
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Which option increases the precision of a confidence interval?
Which option increases the precision of a confidence interval?
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Increasing sample size. Larger sample reduces standard error and interval width.
Increasing sample size. Larger sample reduces standard error and interval width.
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