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.

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

• Examples:

  • Probability that :

  • Probability between two z-scores:

  • Probability to the left of a z-score:

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

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

• Percentiles: To find the value corresponding to a percentile, use the z-score for that percentile and solve for .

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:

• Standard error:

• Example:

  • Population mean = , ,

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 is known):

• Formula (when is unknown):

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

• Examples:

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

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

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

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:

• Always round up to the next whole number.

• Examples:

  • To estimate mean birth weight with grams, , : 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 (): usually states equality (e.g., )

   • Alternative hypothesis (): states inequality (e.g., )

2. Choose the appropriate test:

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

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

3. Calculate the test statistic:

   • z-test:

   • t-test:

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 (), 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:

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

  • z =

  • p-value =

  • At , reject the null hypothesis.

• Testing mean cost of yoga session:

  • Claim:

  • Sample mean = , ,

  • t =

  • Critical value =

  • At , reject the null hypothesis.

• Testing mean MCAT score:

  • Claim:

  • Sample mean = $34s = 3.5n = 75$

  • t =

  • p-value =

  • At , reject the null hypothesis.

• Testing mean hat size:

  • Claim:

  • Sample mean = , ,

  • t =

  • Critical value =

  • At , fail to reject the null hypothesis.

Summary Table: Hypothesis Test Decision Criteria

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.