AP - Statistics:: Inference for Quantitative Data - Means

When constructing a confidence interval for a population MEAN (rather than a proportion), why is the t-DISTRIBUTION used instead of the standard Normal (z) distribution?

- **A)** Because the population standard deviation \(\sigma\) is typically UNKNOWN and must be estimated using the sample standard deviation \(s\), which introduces additional uncertainty that the t-distribution (with its heavier tails) accounts for
- **B)** The t-distribution is used only when the sample size is extremely large
- **C)** The t-distribution is only used for categorical data
- **D)** The t-distribution and z-distribution are always identical

The t-distribution's shape depends on a parameter called DEGREES OF FREEDOM, calculated for a one-sample t-procedure as:

- **A)** \(df = n-1\)
- **B)** \(df = n+1\)
- **C)** \(df = n\)
- **D)** \(df = \sqrt{n}\)

As DEGREES OF FREEDOM increases (i.e., as sample size increases), the t-distribution's shape:

- **A)** Degrees of freedom has no effect on the t-distribution's shape
- **B)** Becomes increasingly different from the Normal distribution
- **C)** Becomes uniform (flat)
- **D)** Becomes increasingly similar to the standard Normal (z) distribution, with thinner tails and less spread

The formula for the ONE-SAMPLE t confidence interval for a population mean is:

- **A)** \(\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}\)
- **B)** \(\mu \pm t^* s\)
- **C)** \(\bar{x} \pm t^* \sigma\)
- **D)** \(\bar{x} \pm z^* \dfrac{\sigma}{\sqrt{n}}\)

The conditions required for valid one-sample t-procedures for a mean include:

- **A)** The population must always be exactly Normal, with no exceptions
- **B)** Only that the sample size is at least 1000
- **C)** No conditions are required for t-procedures
- **D)** RANDOM sample/assignment, the 10% condition (if sampling without replacement), and either a Normal population OR a sufficiently large sample size (often n at least 30, per the CLT) if the population's shape is unknown or skewed

The ONE-SAMPLE t TEST STATISTIC for testing \(H_0: \mu = \mu_0\) is calculated as:

- **A)** \(t = \dfrac{\mu_0}{\bar{x}}\)
- **B)** \(t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}\)
- **C)** \(t = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}\)
- **D)** \(t = \bar{x} - \mu_0\)

A sample of size \(n=25\) has \(\bar{x}=52\), \(s=10\). Test \(H_0: \mu=48\). Find the t test statistic.

- **A)** \(4\)
- **B)** \(2\)
- **C)** \(20\)
- **D)** \(0.4\)

For the t test statistic calculated above (\(t=2\), \(df=24\)), how would you find the associated p-value?

- **A)** The p-value is always exactly equal to the t statistic itself
- **B)** P-values cannot be calculated for t-procedures
- **C)** Use a t-distribution table (or technology) with 24 degrees of freedom to find the area beyond \(t=2\) (in one or both tails, depending on the alternative hypothesis)
- **D)** Use the standard Normal (z) table directly, since t and z distributions are identical