Summarizing Data, Sampling Distributions, Estimation, and Hypothesis Testing

Comprehensive practice flashcards covering data summarization, sampling distributions, large-sample estimation, and statistical hypothesis testing.

How is a dot plot constructed for a numerical variable?

To construct a dot plot, draw a horizontal line, label it with the name of the variable, mark regular values on it, and place a dot above each observation's value on the number line.

What are the four main distribution shapes described when interpreting data graphs?

The four main shapes are mound shaped and symmetric (mirror images), skewed right (a few unusually large measurements), skewed left (a few unusually small measurements), and bimodal (two local peaks).

What is the left inclusion method used when building a histogram table?

It is a tallying method that includes the left endpoint of an interval in the frequency count, but excludes the right endpoint.

What is the key difference between a parameter and a statistic?

A parameter is a numerical descriptive measure calculated for an entire population, whereas a statistic is a numerical descriptive measure calculated for a sample.

How do the mean and median compare in symmetric, right-skewed, and left-skewed distributions?

In symmetric distributions, $\text{mean} = \text{median}$; in right-skewed distributions, $\text{mean} > \text{median}$; and in left-skewed distributions, $\text{mean} < \text{median}$.

What is the sample variance formula $s^2$, and why is the sample standard deviation $s$ preferred for interpreting spread?

The sample variance formula is $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$. Standard deviation ($s = \sqrt{s^2}$) is preferred because taking the positive square root returns the measure of variability to the original units of measurement.

How is the location $L$ calculated for the $p$-th percentile, and what rule is followed if $L$ is not an integer?

The location formula is $L = \frac{n \times p}{100}$. If $L$ is not an integer, round $L$ up to the next largest integer, and the $p$-th percentile is the data value in that position.

Why are the median and the interquartile range ($IQR$) referred to as robust statistics?

They are called robust statistics because extreme observations (outliers) have little to no effect on their values compared to the mean and standard deviation.

How are the lower fence and upper fence calculated to identify outliers in a box plot?

The lower fence is calculated as $Q_1 - 1.5 \times IQR$ and the upper fence is calculated as $Q_3 + 1.5 \times IQR$. Any observation beyond these fences is classified as an outlier.

How do bar plots differ from histograms in terms of variable types and x-axis ordering?

Bar plots display categorical variables and the categories on the x-axis can be listed in any order, whereas histograms display numerical variables where the x-axis is a continuous number line and bar order cannot be changed.

What is a sampling distribution?

A sampling distribution is the probability distribution of a statistic obtained through repeated sampling, describing which values of the statistic can occur and how frequently they occur.

What does the Central Limit Theorem state regarding the sample mean $\bar{x}$?

If random samples of $n$ observations are drawn from a nonnormal population with finite mean $\mu$ and standard deviation $\sigma$, the sampling distribution of $\bar{x}$ is approximately normal with mean $\mu$ and standard error $SE = \frac{\sigma}{\sqrt{n}}$ when $n$ is large ($n \ge 30$).

What is the success-failure condition required for the sampling distribution of a sample proportion $\hat{p}$ to be approximately normal?

The sample size $n$ and population proportion $p$ must satisfy both $n p \ge 10$ and $n(1 - p) \ge 10$ (or $n \hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$ when $p$ is unknown).

What is the formula for the standard error ($SE$) of the sample proportion $\hat{p}$?

The standard error of $\hat{p}$ is $SE = \sqrt{\frac{p(1 - p)}{n}}$.

What properties define an unbiased point estimator with minimum variance?

An estimator is unbiased if the mean of its sampling distribution equals the target parameter. Among unbiased estimators, the preferred estimator is the one with the smallest variability (spread) in its sampling distribution.

What are the $z_{\alpha/2}$ critical values for $90\%$, $95\%$, $98\%$, and $99\%$ confidence intervals?

The critical values are $z_{\alpha/2} = 1.645$ for $90\%$, $z_{\alpha/2} = 1.96$ for $95\%$, $z_{\alpha/2} = 2.33$ for $98\%$, and $z_{\alpha/2} = 2.58$ for $99\%$ confidence levels.

What is the formula for a large-sample $100(1 - \alpha)\%$ confidence interval for a population mean $\mu$?

The confidence interval formula is $\bar{x} \pm z_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)$, where $z_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)$ represents the margin of error.

What is the formula for a large-sample confidence interval for the difference between two population means $\mu_1 - \mu_2$?

The confidence interval formula is $(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.

How is the inclusion of zero in a confidence interval for $\mu_1 - \mu_2$ or $p_1 - p_2$ interpreted?

If 0 is included inside the confidence interval, it is possible that the two population parameters are equal, so no difference should be declared. If 0 is not in the interval, it is unlikely they are equal, and a statistically significant difference can be declared.

What are the roles of the null hypothesis $H_0$ and the alternative hypothesis $H_a$ in statistical testing?

The null hypothesis ($H_0$) is assumed to be true until sample evidence proves otherwise, while the alternative hypothesis ($H_a$) is the claim accepted as true if $H_0$ is rejected.

What is the difference between a Type I error and a Type II error?

A Type I error occurs when rejecting $H_0$ when $H_0$ is actually true (with probability $\alpha$). A Type II error occurs when failing to reject $H_0$ when $H_0$ is actually false (with probability $\beta$).

What is the test statistic formula $z$ for testing a population mean $\mu$ against a null value $\mu_0$ in a large sample?

The test statistic is $z = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$.

What is the test statistic formula $z$ for testing a binomial proportion $p$ against a null value $p_0$?

The test statistic is $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$, used when $n p_0 \ge 10$ and $n(1 - p_0) \ge 10$.

What is the definition of a $p$-value in hypothesis testing?

The $p$-value is the probability of observing, just by chance, a test statistic as extreme as or more extreme than the observed value, assuming the null hypothesis $H_0$ is true.

According to general statistical guidelines, how are results categorized based on their $p$-value when no significance level $\alpha$ is specified?

If $p < 0.01$, results are highly significant; if $0.01 \le p \le 0.05$, results are statistically significant; if $0.05 < p \le 0.10$, results tend towards significance; and if $p > 0.10$, results are not statistically significant.