Statistics Practice Test Study Guide: Probability, Normal Distribution, Confidence Intervals, and Hypothesis Testing

Probability and the Normal Distribution

Finding Probabilities Using the Standard Normal Distribution

The standard normal distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1. Probabilities for normal distributions are often found using z-scores and standard normal tables.

• Z-score: The number of standard deviations a value is from the mean. $z = \frac{x - \mu}{\sigma}$

• Finding probabilities: Use the z-score to look up the area (probability) in the standard normal table.

• Examples:

  • Probability that $z > 2.15$: $P(z > 2.15) = 0.016$

  • Probability between two z-scores: $P(-2.37 < z < -0.37) = 0.0516$

  • Probability to the left of a z-score: $P(z < 0.37) = 0.6440$

Applications of the Normal Distribution

Many real-world variables (e.g., utility bills, braking distances, pay scales) are modeled using the normal distribution. Probabilities and percentiles can be calculated using the mean and standard deviation.

• Example:

  • Monthly utility bills: mean = $100$, standard deviation = $12$

  • Probability bill is less than $70$:$z = \frac{70 - 100}{12} = -2.5P(x < 70) = P(z < -2.5) = 0.0062$

• Percentiles: To find the value corresponding to a percentile, use the z-score for that percentile and solve for $x$. $x = \mu + z\sigma$

Sampling Distributions and Confidence Intervals

Sampling Distribution of the Mean

The sampling distribution of the sample mean describes the distribution of means from all possible samples of a given size from a population.

• Mean of sampling distribution: $\mu_{\bar{x}} = \mu$

• Standard error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Example:

  • Population mean = $19.5$, $\sigma = 6.16$, $n = 40$ $\sigma_{\bar{x}} = \frac{6.16}{\sqrt{40}} = 0.974$

Confidence Intervals for the Mean

A confidence interval estimates the range in which the population mean is likely to fall, based on sample data.

• Formula (when $\sigma$ is known): $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

• Formula (when $\sigma$ is unknown): $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

• Interpretation: "With 95% confidence, the population mean is between [lower bound] and [upper bound]."

• Examples:

  • College student earnings: 95% CI is ($52803.16$, $34346.84$)

  • TV hours for eighth graders: 90% CI is ($18.31$, $20.89$)

  • Women's mean height: 99% CI is ($61.45$, $65.35$)

Effect of Standard Deviation and Sample Size on Confidence Intervals

The width of a confidence interval depends on the standard deviation and sample size.

• Increasing standard deviation: Widens the confidence interval (less precision).

• Increasing sample size: Narrows the confidence interval (more precision).

Sample Size Determination

To estimate a population mean with a specified margin of error and confidence level, use the following formula:

• Formula: $n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$

• Always round up to the next whole number.

• Examples:

  • To estimate mean birth weight with $E = 0.2$ grams, $\sigma = 0.7$, $z_{\alpha/2} = 2.575$: $n = \left( \frac{2.575 \cdot 0.7}{0.2} \right)^2 = 81.225$ Need to sample 82 infants.

Hypothesis Testing for One Sample Mean

Steps in Hypothesis Testing

Hypothesis testing is used to determine if there is enough evidence to support a claim about a population parameter.

1. State the null and alternative hypotheses:

   • Null hypothesis ($H_0$): usually states equality (e.g., $\mu = 135$)

   • Alternative hypothesis ($H_a$): states inequality (e.g., $\mu < 135$)

2. Choose the appropriate test:

   • Use z-test if $\sigma$ is known and sample size is large.

   • Use t-test if $\sigma$ is unknown and/or sample size is small.

3. Calculate the test statistic:

   • z-test: $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

   • t-test: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

4. Find the p-value or critical value:

   • Compare the test statistic to the critical value or use the p-value approach.

5. Make a decision:

   • If p-value $<$ significance level ($\alpha$), reject the null hypothesis.

   • If p-value $>$ significance level, fail to reject the null hypothesis.

6. Interpret the results in context.

Examples of Hypothesis Tests

• Testing mean activating temperature:

  • Claim: $\mu \geq 135$

  • Sample mean = $133$,$\sigma = 3.3$,$n = 22$

  • z = $\frac{133 - 135}{3.3 / \sqrt{22}} = -3.43$

  • p-value = $0.0003$

  • At $\alpha = 0.10$, reject the null hypothesis.

• Testing mean cost of yoga session:

  • Claim: $\mu \leq 14$

  • Sample mean = $15.59$, $s = 2.60$, $n = 29$

  • t = $3.293$

  • Critical value = $2.048$

  • At $\alpha = 0.025$, reject the null hypothesis.

• Testing mean MCAT score:

  • Claim: $\mu > 33$

  • Sample mean = $34$,$s = 3.5$,$n = 75$

  • t = $2.474$

  • p-value = $0.0068$

  • At $\alpha = 0.05$, reject the null hypothesis.

• Testing mean hat size:

  • Claim: $\mu \geq 7.25$

  • Sample mean = $7.15$, $s = 0.27$, $n = 12$

  • t = $-1.283$

  • Critical value = $-1.796$

  • At $\alpha = 0.05$, fail to reject the null hypothesis.

Summary Table: Hypothesis Test Decision Criteria

Test Statistic	Critical Value	p-value	Decision
z or t	Depends on test and significance level	Calculated from test statistic	Reject $H_0$ if p-value $<$ $\alpha$
Example: z = -3.43	-1.645 (for $\alpha = 0.05$)	0.0003	Reject $H_0$
Example: t = 1.642	1.740 (for $\alpha = 0.10$)	0.10	Fail to reject $H_0$

Additional info:

• Some problems involve determining whether to use a normal or t-distribution based on whether the population standard deviation is known and the sample size.

• Interpretation of confidence intervals and hypothesis test results is emphasized throughout.

• Sample size calculations always require rounding up to the next whole number.