AP Statistics Flashcards: Confidence Interval For A Population Mean

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This deck focuses on Confidence Interval For A Population Mean, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What happens to the interval if sample standard deviation decreases?

Answer: The interval becomes narrower. Smaller $s$ reduces standard error and margin of error.

Flashcard 2: How does variability affect the confidence interval?

Answer: Greater variability widens the interval. Higher $s$ increases standard error and interval width.

Flashcard 3: Find $t^*$ for $df = 10$ at 95% confidence level.

Answer: Approximately 2.228. From $t$-table with $df = 10$ and 95% confidence level.

Flashcard 4: Find the 95% confidence interval: $\bar{x} = 10$, $t^* = 2$, $s = 5$, $n = 25$.

Answer: $10 \pm 2$. Margin of error is $2 \times \frac{5}{\sqrt{25}} = 2 \times 1 = 2$.

Flashcard 5: Find $t^*$ for $df = 10$ at 95% confidence level.

Answer: Approximately 2.228. From $t$-table with $df = 10$ and 95% confidence level.

Flashcard 6: Identify the conditions for normality in constructing confidence intervals.

Answer: Sample size large or population distribution approximately normal. Ensures sampling distribution of $\bar{x}$ is approximately normal.

Flashcard 7: Which option increases the precision of a confidence interval?

Answer: Increasing sample size. Larger sample reduces standard error and interval width.

Flashcard 8: Find the degrees of freedom for $n = 15$.

Answer:

1. Always $n - 1$ for single-sample $t$-procedures.

Flashcard 9: What happens to a confidence interval if the confidence level decreases?

Answer: The interval becomes narrower. Lower confidence level uses smaller critical value.

Flashcard 10: What is the central limit theorem in context of confidence intervals?

Answer: The sampling distribution of the sample mean is approximately normal. Allows use of normal approximation for sample means.

Flashcard 11: Identify the degrees of freedom for a sample size of 25.

Answer:

1. Degrees of freedom equals $n - 1 = 25 - 1 = 24$.

Flashcard 12: What does a 95% confidence level imply?

Answer: 95% of such intervals will capture the true population mean. Long-run frequency of intervals containing true parameter.

Flashcard 13: State the formula for a confidence interval for a mean.

Answer: $\bar{x} \, \pm \, t^* \left(\frac{s}{\sqrt{n}}\right)$. Sample mean plus/minus margin of error using $t$-distribution.

Flashcard 14: What is the relationship between confidence level and margin of error?

Answer: Higher confidence increases margin of error. Higher confidence requires larger critical value and margin.

Flashcard 15: State the formula for a confidence interval for a mean.

Answer: $\bar{x} \, \pm \, t^{*} \left(\frac{s}{\sqrt{n}}\right)$. Sample mean plus/minus margin of error using $t$-distribution.

Flashcard 16: Which table do you use to find $t^*$ values?

Answer: The $t$-distribution table. Critical values depend on confidence level and degrees of freedom.

Flashcard 17: Find the 95% confidence interval: $\bar{x} = 10$, $t^* = 2$, $s = 5$, $n = 25$.

Answer: $10 \pm 2$. Margin of error is $2 \times \frac{5}{\sqrt{25}} = 2 \times 1 = 2$.

Flashcard 18: How does variability affect the confidence interval?

Answer: Greater variability widens the interval. Higher $s$ increases standard error and interval width.

Flashcard 19: Identify the effect of decreasing variability on the confidence interval.

Answer: It narrows the interval. Less variability reduces standard error and margin of error.

Flashcard 20: Calculate the standard error: $s = 3$, $n = 36$.

Answer: $\frac{3}{\sqrt{36}} = 0.5$. Standard error formula: $s$ divided by square root of $n$.

Flashcard 21: Find the margin of error given $t^* = 2$, $s = 4$, $n = 16$.

Answer: $2 \times \frac{4}{\sqrt{16}} = 2$. Margin of error equals $t^* \times$ standard error.

Flashcard 22: What does $t^*$ represent in the confidence interval formula?

Answer: Critical value from the $t$-distribution. Based on confidence level and degrees of freedom.

Flashcard 23: State the formula for standard error of the mean.

Answer: $\frac{s}{\sqrt{n}}$. Standard deviation of the sampling distribution of $\bar{x}$.

Flashcard 24: What is the relationship between confidence level and margin of error?

Answer: Higher confidence increases margin of error. Higher confidence requires larger critical value and margin.

Flashcard 25: What does $\bar{x}$ represent in the confidence interval formula?

Answer: Sample mean. The average of all observations in the sample.

Flashcard 26: State the formula for standard error of the mean.

Answer: $\frac{s}{\sqrt{n}}$. Standard deviation of the sampling distribution of $\bar{x}$.

Flashcard 27: What is the effect of confidence level on interval width?

Answer: Higher confidence level widens the interval. More confidence requires wider interval to capture parameter.

Flashcard 28: Calculate the standard error: $s = 3$, $n = 36$.

Answer: $\frac{3}{\sqrt{36}} = 0.5$. Standard error formula: $s$ divided by square root of $n$.

Flashcard 29: Calculate confidence interval: $\bar{x} = 20$, $t^* = 1.96$, $s = 4$, $n = 100$.

Answer: $20 \pm 0.784$. Margin of error: $1.96 \times \frac{4}{\sqrt{100}} = 1.96 \times 0.4 = 0.784$.

Flashcard 30: What is the effect of confidence level on interval width?

Answer: Higher confidence level widens the interval. More confidence requires wider interval to capture parameter.

Flashcard 31: What does $\bar{x}$ represent in the confidence interval formula?

Answer: Sample mean. The average of all observations in the sample.

Flashcard 32: How do you interpret a confidence interval?

Answer: A range where the population mean is likely to be found. Plausible values for the unknown population mean.

Flashcard 33: Identify the condition for using a $t$-distribution in confidence intervals.

Answer: Population standard deviation is unknown and sample is small. Use when $\sigma$ is unknown, especially with small samples.

Flashcard 34: Which distribution is used when population standard deviation is known?

Answer: Normal distribution (Z-distribution). Known $\sigma$ allows use of standard normal distribution.

Flashcard 35: Identify the sample mean in the data set: 5, 6, 7, 8, 9.

Answer: $\bar{x} = 7$. Sum of values divided by count: $(5+6+7+8+9)/5 = 7$.

Flashcard 36: How does sample size relate to the standard error?

Answer: Larger sample size decreases standard error. Standard error decreases as $n$ increases: $\frac{s}{\sqrt{n}}$.

Flashcard 37: Calculate the interval: $\bar{x} = 50$, $t^* = 2.5$, $s = 10$, $n = 16$.

Answer: $50 \pm 6.25$. Margin of error: $2.5 \times \frac{10}{\sqrt{16}} = 2.5 \times 2.5 = 6.25$.

Flashcard 38: Which term describes the range of values in a confidence interval?

Answer: Margin of error. Half-width of confidence interval around point estimate.

Flashcard 39: What does $s$ represent in the confidence interval formula?

Answer: Sample standard deviation. Measures spread of sample observations around sample mean.

Flashcard 40: Which distribution is used when population standard deviation is known?

Answer: Normal distribution (Z-distribution). Known $\sigma$ allows use of standard normal distribution.

Flashcard 41: Which factor has no effect on interval width if other factors are fixed?

Answer: Sample mean. Sample mean is center point; doesn't affect interval width.

Flashcard 42: Find the margin of error given $t^* = 2$, $s = 4$, $n = 16$.

Answer: $2 \times \frac{4}{\sqrt{16}} = 2$. Margin of error equals $t^* \times$ standard error.

Flashcard 43: What does $n$ represent in the confidence interval formula?

Answer: Sample size. Total number of observations in the sample.

Flashcard 44: What does a 95% confidence level imply?

Answer: 95% of such intervals will capture the true population mean. Long-run frequency of intervals containing true parameter.

Flashcard 45: Identify the conditions for normality in constructing confidence intervals.

Answer: Sample size large or population distribution approximately normal. Ensures sampling distribution of $\bar{x}$ is approximately normal.

Flashcard 46: Identify the effect of increasing sample size on the confidence interval.

Answer: It narrows the confidence interval. Larger $n$ reduces standard error and margin of error.

Flashcard 47: What is the central limit theorem in context of confidence intervals?

Answer: The sampling distribution of the sample mean is approximately normal. Allows use of normal approximation for sample means.

Flashcard 48: Identify the effect of decreasing variability on the confidence interval.

Answer: It narrows the interval. Less variability reduces standard error and margin of error.

Flashcard 49: Which factor has no effect on interval width if other factors are fixed?

Answer: Sample mean. Sample mean is center point; doesn't affect interval width.

Flashcard 50: Identify the sample mean in the data set: 5, 6, 7, 8, 9.

Answer: $\bar{x} = 7$. Sum of values divided by count: $(5+6+7+8+9)/5 = 7$.

Flashcard 51: Find the degrees of freedom for $n = 15$.

Answer:

1. Always $n - 1$ for single-sample $t$-procedures.

Flashcard 52: What does a wider confidence interval suggest about precision?

Answer: Less precision. Wider intervals indicate greater uncertainty about parameter.

Flashcard 53: What does $s$ represent in the confidence interval formula?

Answer: Sample standard deviation. Measures spread of sample observations around sample mean.

Flashcard 54: Identify the degrees of freedom for a sample size of 25.

Answer:

1. Degrees of freedom equals $n - 1 = 25 - 1 = 24$.

Flashcard 55: What is the primary purpose of a confidence interval?

Answer: To estimate a population parameter. Uses sample data to infer about population parameters.