Justifying Claims: Population Mean, Confidence Interval - AP Statistics

2.576. Critical value leaving 0.5% in each tail of standard normal distribution.

Calculate the sample mean, $\bar{x}$. Point estimate serves as center of the confidence interval.

Data must be approximately normal if $n < 30$. Small samples require normality assumption for valid inference.

$\bar{x} \text{ ± } z^* \frac{s}{\text{√}n}$. Standard formula: sample mean plus/minus critical value times standard error.

Larger samples increase reliability. Larger samples reduce sampling variability and increase precision.

Possible lack of effect or no difference. Zero in interval suggests no significant difference from zero.

Sample mean, $\bar{x}$. Center of symmetric interval equals the point estimate.

Reduces margin of error. Standard error decreases with $\sqrt{n}$, reducing uncertainty.

When assumptions about sample and distribution hold. Requires proper sampling method and satisfied distributional assumptions.

45 to 55. Sample mean ± margin of error gives the interval bounds.

$z^* = 1.96$. Standard value for 95% confidence from normal distribution table.

Narrower interval. Smaller critical value decreases the margin of error.

Wider interval. Larger critical value needed increases the margin of error.

1. Degrees of freedom equals $n - 1$ for one-sample $t$-procedures.

$z^* \frac{s}{\text{√}n}$. Measures uncertainty in the estimate based on sample variability.

1. Midpoint of interval endpoints: $(10 + 20)/2 = 15$.

Sample mean. Point estimate for the population mean from the sample data.

When population standard deviation is unknown and $n < 30$. Accounts for additional uncertainty when $\sigma$ is unknown with small samples.

Approximately 2.262. From $t$-table with $df = 9$ and $\alpha/2 = 0.025$.

Sample size and variability. Larger $n$ reduces width; higher variability increases width.

It halves. Standard error is inversely proportional to $\sqrt{n}$.

The width decreases. Standard error decreases as $n$ increases, reducing margin of error.

The width increases. Higher confidence requires larger critical value, increasing margin of error.

Increases the width. Higher $s$ increases standard error and thus margin of error.

To estimate a population parameter. Provides range of plausible values for unknown population parameter.

When assumptions about sample and distribution hold. Requires proper sampling method and satisfied distributional assumptions.

Larger samples increase reliability. Larger samples reduce sampling variability and increase precision.

Possible lack of effect or no difference. Zero in interval suggests no significant difference from zero.

Sample mean, $\bar{x}$. Center of symmetric interval equals the point estimate.