Sampling Distributions: Normal Distribution

Introduction to Sampling Distributions

• Unit 5 focuses on sampling distributions, a critical link between prior knowledge and inference.
• The unit connects previously learned concepts to what will be learned next.
• While a review, the video doesn't cover every detail but focuses on key concepts.
• A study guide is recommended for practice.

Normal Distribution Revisited

• The normal distribution can model continuous random variables.
• A continuous random variable can take any numerical value within a specified domain.
• Continuous random variables have probabilities associated with intervals within their domain.
• Probability of an individual specific numerical outcome cannot be found, unlike discrete random variables.
• The normal distribution is defined by its mean and standard deviation.
• 99.7% of outcomes fall within three standard deviations of the mean (negative three to positive three)
• The area between two points on a normal distribution represents the probability of an outcome falling within that interval.
• Probabilities can be found using a z-table or technology like the normalcdf feature on a TI-84 calculator.

Example 1: Maxi's Savings

• Maxi's monthly savings follow a normal distribution with a mean of $55.20 and a standard deviation of $8.15.
• To find the probability that Maxi contributes more than $60 next month, calculate the z-score for $60.
• The z-score is calculated as (60 - 55.20) / 8.15 = 0.589.
• The probability of contributing more than $60 is equivalent to the probability of a z-score greater than 0.589.
• There's a 27.8% probability that Maxi contributes more than $60 in a given month, based on the z-score.

Example 2: Top and Bottom Contributions

• To find the amounts representing the top and bottom 5% of contributions, find the corresponding z-scores.
• The z-scores for the top and bottom 5% are 1.645 and -1.645, respectively.
• These z-scores can be found using a z-table in reverse OR the invNorm function on a TI-84 calculator.
• Plug the z-scores into the z-score formula and solve for x, giving the amount of money.
• $68.61+ represents the top 5% of contributions.
• $41.79- represents the bottom 5% of contributions.

Example 3: Combined Savings

• Cassandra's average contribution is $62.45 with a standard deviation of $12.66.
• To find the probability that the total saved by Maxi and Cassandra is greater than $140, combine their distributions.
• The mean of the total contributions is the sum of their individual means: $55.20 + $62.45 = $117.65.
• Variances can be legally combined, not standard deviations.
• Combined variance = (8.15)^2 + (12.66)^2.
• Take the square root of the combined variance to find the standard deviation, ≈ $15.056.
• To find the probability that the total exceeds $140: calculate the z-score for $140 ≈ 1.484.
• The probability of a z-score exceeding 1.484 is 6.89%.

Samples and Population Parameters

• Samples are taken and analyzed to estimate population parameters.
• A sample statistic summarizes information about a sample.
• The goal of collecting a sample statistic is to infer something about the population parameter.
• A sample statistic is a point estimator of the population parameter.
• A sample mean (x̄) is a point estimate of the population mean (μ).
• A sample proportion (p̂) estimates the population proportion (p).
• Sample statistics will not perfectly match population parameters due to sampling variability.
• Sampling variability recognizes that samples vary amongst themselves and from the population parameter.

Repeated Sampling

• Repeated samples of the same size are taken from the same population.
• Samples must be random to avoid bias.
• Samples must be independent of each other.
• Independence is guaranteed through sampling with replacement.
• When sampling without replacement, the sample size (n) must be less than 10% of the population size.
• If n < 10% of the population, then all of the sampling distribution rules covered will apply.

Sampling and Independence

• When sampling without replacement, samples are not independent if a significant portion of the population is removed.
• Removing a small amount (under 10%) of the population makes the lack of independence negligible.
• The 10% condition allows for the assumption of independence between samples.
• Meeting the random samples and 10% population rule enables sampling.

Sampling Distributions

• A sampling distribution is a distribution of values for a statistic from all possible samples of a given size from a given population.
• When analyzing a quantitative variable, sample means from multiple samples form a distribution.
• This distribution of sample means is a sampling distribution.

Sampling Distributions and Categorical Variables

• For categorical variables, sample proportions (P-hats) are collected from multiple samples.
• Each P-hat is a quantitative variable.
• The distribution of all P-hats forms a sampling distribution for a sample proportion.

Sampling Distributions for Other Statistics

• Sampling distributions can be created for medians by collecting medians from multiple samples.
• Sampling distributions can also be created for ranges by collecting ranges from multiple samples.
• A sampling distribution is a collection of all possible sample statistics for all possible samples of a given sample size from a given population.
• The two most popular sample statistics are the sample mean (X-bar) and the sample proportion (P-hat).

Sampling Distribution Example: Sample Proportions

• In a city, 65% of registered voters plan to vote "Yes" on an issue.
• Repeated random samples of size 150 are taken.
• It's assumed that 150 voters are under 10% of all registered voters in the city allowing for an assumption of independence.
• Collecting the sample proportion of "Yes" voters from each sample yields a bunch of P-hats.
• When these P-hats are put together, they create a distribution that are the start of the sampling distribution.
• Each green dot in the example represents an individual sample proportion.
• Samples vary, which is called sampling variability.

Characteristics of Sampling Distributions

• Center: The mean of all P-hats is approximately 65% (0.65).
• Sample statistics that point to the truth are unbiased estimators.
• If the center is not 0.65, the sample estimate would be biased.
• Spread: There is variability among the samples due to sampling variability.
• As a collective group, the mean of all samples aligns with the truth at the center.
• Shape: The shape of the sampling distribution is normal.

Sampling Distribution Example: Sample Means

• Cell phone true mean weight: The true mean weight of all cell phones at a high school (including the case) is 180 grams.
• Repeated samples of 45 cell phones are taken.
• From the cell phones samples the mean weight of those cell phones are analysed, and more samples are taken.
• Creating a distribution with all the sample means (X-bars) creates the start of the sampling distribution.
• Sampling distribution center: The mean of all sample means piles up around the true mean of 180 grams.
• The sample estimates are unbiased
• The sample means something is biased if they pile up around 160.
• Sampling distribution spread: Not every sample is 180 grams; there is variability.
• Sampling distribution shape: Approximates normal.

Central Limit Theorem

• The Central Limit Theorem (CLT) states that when the sample size is sufficiently large (at least 30), a sampling distribution of the mean will be approximately normal.
• Regardless of the population distribution, the sampling distribution of the mean will be normal if the sample size is 30 or larger.

Modeling Sampling Distributions

• Simulating a thousand sample statistics is not a true sampling distribution.
• A true sampling distribution requires values for all possible statistics from all possible samples.
• Creating a model of what the sampling distribution could look like is an alternative.
• To build a model of a sampling distribution, the population parameters need to be known.

Sampling Distribution Model Requirements

• Center or mean of all possible P-hats should equal the rue population parameter p
• Samples must be random to avoid bias.
• Spread is the standard deviation of all P-hats can be found by the square root of P times 1 minus P all divided by the sample size n.
• Samples must be less than 10% of the population to assume independence.
• Shape: It has to be normal as long as samples are big enough.
• Sample needs 10+ successes and 10+ failures expected.

Sampling Distribution Model Example

• True proportion of voters voting yes is 65%.
• Need the center: Mean of all P-hats is 0.65, this sample is unbiased.
• Need the spread: Standard deviation is square root of 0.65 * 0.35 / sample size of 150 = 0.0389. Assume sample independence
• Need the shape: Should be normal, with 10+ of successes and failures with this calculation: (150 * 0.65) and (150 * 0.35).

Variability Measurement in Model

• This measurement allows to see what possible the P-hats could look like.
• Model means that can examine the variation within the model using probability (Ex: Sample proportion with 150 registered voters is less than 58%?)
• In order to get an answer, need a created model and need to identify what a sample proportion is with 58, so .58 to a z-score of -1.799 (-1.80 per instructions)
• Z-score less than that (-1.79), can use calculator or Z tables.
• Can also ask questions like "what samplemarks the top 5% of all sample proportions?
• Can use calculator to find the z-score of the top 5% by using invnorm. The top 5% is equivalent to the bottom 95%. Answer = 1.645
• Use the z-score formula, and use variables we do know (1.545 score, 0.65 mean, 0.039).
• Calculate: Multiply the standard deviation, add the mean, P-hat = 0.714.
• If a sample proportion returns 71.4 or higher is in the top 5% of al possible sample proportions.

Sampling Distribution for the Difference

• It's sample proportions between two different populations, population 1 v population 2.

Sampling Distribution for Differences in Sample Proportions

• The sampling distribution for differences in sample proportions requires:
  • The center
  • The spread
  • The shape

Center

• The center is the mean of all possible differences between sample proportions from two populations.
• This mean should equal the true difference between the parameters of the two populations.
• This is valid only if both samples are random to avoid bias.

Spread

• The spread is the standard deviation of all possible differences between sample proportions from two populations.
• It is calculated using a formula available on the AP stats form sheet.
• The samples must be independent.
• Independence is assumed if sample sizes are less than 10% of their respective populations.
• Independence must also exist between the two samples themselves.

Shape

• The shape is normal if both samples are large enough.
• "Large enough" means each sample has at least 10 successes and 10 failures.

Example: School Districts

• District A has 80% of students passing a math test, while District B has 76%.
• The goal is to model differences between sample proportions, with 75 students from District A and 100 from District B.
• Center: with a true difference of 4% (0.04), based on the difference in population proportions.
• Standard Deviation: Calculated using the success and failure rates from both districts and their sample sizes, resulting in 0.0629.
• Shape: Confirmed to be normal because both samples have at least 10 successes and failures.

Probability Calculation

• The probability that a sample from District B has a higher proportion of passing students than a sample from District A can be modeled.
• A negative difference indicates that the proportion from District B is higher than District A.
• To find this probability, calculate the z-score for a zero difference
• Determine the area below this z-score using a normal distribution.
• A z-score of -0.636 corresponds to a 26.2% chance that the sample proportion from District B is higher than District A.

Sampling Distribution for Sample Means

• For a numerical variable, a sampling distribution models the distribution of all possible sample means
• Requires:
  • The population mean (μ)
  • The population standard deviation (σ)
  • The sample size (n)

Center

• The center is the mean of all sample means, which should equal the true population mean.
• This assumes samples are random to avoid bias.

Spread

• The spread is the standard deviation of the sampling distribution.
• Calculated as the population standard deviation (σ) divided by the square root of the sample size (n).
• Samples must be independent, assumed if the sample size is less than 10% of the population.

Shape

• The shape can be normal under two conditions:
  • If the population is normally distributed, the sampling distribution will be normal regardless of sample size.
  • If the population is not normally distributed, the Central Limit Theorem states that the sampling distribution will be approximately normal if the sample size is 30 or larger.

Example: Cell Phone Weights

• Cell phones at Roosevelt High School have a mean weight of 180 grams and a standard deviation of 15 grams.
• The population distribution is skewed to the right.
• Sample size: 45 cell phones
• Center: 180 grams, the true population mean.
• Standard Deviation: 15 / √45 = 2.236 grams.
• Shape: Normal, due to the Central Limit Theorem, since the sample size is greater than 30.

Probability Calculation

• To find the probability that a sample of 45 cell phones has a mean weight greater than 182.5 grams, calculate the z-score.
• Use the z-score to find the probability using a normal distribution.
• A z-score of 1.118 corresponds to a probability of 0.132.

Interval Creation

• To create an interval for the middle 95% of all sample means
• Find the z-scores that bound the middle 95% of the distribution (e.g., -1.96 and 1.96).
• Convert these z-scores back to sample mean values using the z-score formula.
• Any sample mean within this interval represents the middle 95% of all possible sample means.
• The middle 95% of sample means falls between 175.62 grams and 184.38 grams.

Sampling Distribution for Differences in Sample Means

• Examines the possible differences between sample means from two populations.
• Requires:
  • The center
  • The spread
  • The shape

Center

• The center is the mean of all possible differences between sample means from two populations.
• This should equal the true difference between the population means.
• Samples must be random to avoid bias.

Spread

• The spread is the standard deviation of all possible differences between sample means.
• Samples must be independent, which is assumed if both sample sizes are less than 10% of their respective populations.

Shape

• The shape is normal if:
  • Both populations are normally distributed and any sample size will suffice
  • Populations are non-normal, the Central Limit Theorem applies if both sample sizes are 30 or larger.

Example: Orca Whale Weights

• Mean weight of 15 male orca whales and 10 female orca whales
  • Male orcas: normal distribution, mean of 12,000 pounds
  • Standard deviation of 800 pounds.
  • Female orcas: normal distribution, mean of 10,000 pounds
  • Standard deviation of 900 pounds.
• Center: with an expected difference of 2,000 pounds.
• Standard Deviation equals 351.663 pounds
• Shape: Normal because the populations is normal.

Sampling Distribution Model

• The sampling distribution model is sensitive to 2,000 pounds, allowing for slight variations.

Probability of Sample Mean Difference (Orcas)

• Task: Determine the probability that a sample of 15 male orcas has a mean weight 3,000 pounds greater than a sample of 10 female orcas.
• A 3,000-pound difference is located on the sampling distribution.
• The corresponding z-score is 2.844.
• The probability of the mean difference being greater than 3,000 pounds is equivalent to finding the probability of a z-score exceeding 2.844.
• Results: The calculated probability is 0.00223, indicating a very unlikely event.

Unlikely Events and Questioning Original Information

• In statistics, low probability events prompt a re-evaluation of initial assumptions.
• If a sample of male orcas averages 3,000 pounds more than females, it challenges the original average weight data for male and female orcas (12,000 and 10,000 pounds, respectively).
• The focus shifts to questioning the accuracy of the original information provided.

Impact of Sample Size on Standard Deviation

• Sample size (n) is in the denominator of standard deviation formulas.
• Larger sample sizes lead to smaller standard deviations.
• Bigger samples vary less, providing more reliable estimates of population parameters.
• Estimating the average weight of bullfrogs with a sample of 2,000 is more accurate than with a sample of just two.

Example: Cell Phone Weights

• Standard deviation of cell phone weights is 15.
• With a sample size of 45, the standard deviation is 15 / √45 = 2.236.
• Increasing the sample size to 100 reduces the standard deviation to 15 / √100 = 1.5.
• Larger sample sizes yield more reliable values closer to the true population parameter.

Sampling Distribution Comparison

• Two sampling distributions are compared, one in blue and one in red.
• The distribution in blue is narrower, indicating less spread and higher accuracy.
• Both distributions are centered at 180 grams, representing the true value.
• A larger sample size of 100 results in a narrower, more reliable distribution.

Unit 5 Summary

• Four different sampling distributions were explained, each with a center, spread, and shape.
• Formulas for these aspects are available on AP Statistics formula sheets, categorized by proportions and means.
• Understanding what a sampling distribution is is essential.
• Determine how to find its center, spread, and shape, based on population parameters and sample size.

Conditions and Potential Issues

• Conditions must be checked to ensure normality and validity of standard deviation.
• Failed conditions may lead to non-normality, unusable standard deviations, and potential bias.

Probability Questions and Distributions

• Using normal distributions to ask and answer probability questions is emphasized.
• This skill is important for the AP exam.
• Review the study guide for different problem types and how to build sampling distributions and calculate probabilities.