Data Analysis for Marketing Decisions: Session 2
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Questions and Answers

What is the empirical rule that characterizes normal distributions?

  • 60-80-95 rule
  • 90-95-99 rule
  • 80-90-95 rule
  • 68-95-99 rule (correct)

In a standard normal distribution, what is the mean?

  • 0.5
  • It varies
  • 1
  • 0 (correct)

What does the symbol SD represent in the context of normal distributions?

  • Simple difference
  • Standard deviation (correct)
  • Sample data
  • Specific distribution

What proportion of the data falls below the mean in a normal distribution?

<p>Exactly 50% (C)</p> Signup and view all the answers

If a student scores below 58 in a normal distribution, what does it indicate about their performance?

<p>They performed below average (C)</p> Signup and view all the answers

Which statement is true regarding normal distributions?

<p>All normal distributions are symmetric about the mean. (D)</p> Signup and view all the answers

In a probability distribution, what does a higher standard deviation imply?

<p>Greater variability in data (A)</p> Signup and view all the answers

What is the relationship between mean, median, and mode in a normal distribution?

<p>Mean = Median = Mode (B)</p> Signup and view all the answers

What is the primary purpose of statistical inference in data analysis?

<p>To analyze sample data (A)</p> Signup and view all the answers

Which element is NOT part of statistical inference?

<p>Conducting an experimental study (A)</p> Signup and view all the answers

What is a probability distribution?

<p>A function describing the likelihood of different values of a random variable (A)</p> Signup and view all the answers

Why is it important to obtain a random sample in statistical inference?

<p>To minimize bias in estimates (A)</p> Signup and view all the answers

What step follows the summarization of sample data in the statistical inference process?

<p>Modeling the hypothesis using a test statistic (D)</p> Signup and view all the answers

What is a commonly used method of statistical testing in hypothesis formulation?

<p>Null hypothesis significance testing (C)</p> Signup and view all the answers

Which term describes the characteristics of a specific population being inferred in statistical analysis?

<p>Parameters (B)</p> Signup and view all the answers

In the context of statistical inference, which of the following represents contrasts and comparisons?

<p>Comparing sample means to population means (B)</p> Signup and view all the answers

What does the mean of the sample represent in relation to the population?

<p>It is an estimate of the corresponding population parameter. (A)</p> Signup and view all the answers

What does a sampling error imply when using a sample to infer about a population?

<p>The sample statistic may differ from the population parameter. (B)</p> Signup and view all the answers

Given a mean of 55 and a standard deviation of 5, what is the probability of observing a grade lower than 58?

<p>Approximately 73% (A)</p> Signup and view all the answers

What characterizes a hypothesis in scientific research?

<p>It can be tested and potentially disproved. (A)</p> Signup and view all the answers

Which of the following best describes parameter estimation?

<p>It involves collecting data to estimate population parameters. (A)</p> Signup and view all the answers

The 27% mentioned in the context of grades indicates what?

<p>The percentage of grades equal to or higher than 58. (C)</p> Signup and view all the answers

In the context of a hypothesis, what does the term 'dependent variable' refer to?

<p>The outcome that is measured in response to changes. (D)</p> Signup and view all the answers

What is the role of a confidence level in statistical inference?

<p>It indicates the likelihood that the sample accurately represents the population. (D)</p> Signup and view all the answers

What is the difference between a directional hypothesis and a non-directional hypothesis?

<p>Directional hypotheses state an expected direction, while non-directional do not. (A)</p> Signup and view all the answers

What type of statistical test is associated with directional hypotheses?

<p>One-tailed test (C)</p> Signup and view all the answers

How can one calculate the sampling error given a level of confidence?

<p>By using confidence intervals with the sample statistic. (A)</p> Signup and view all the answers

Which of the following statements is true regarding probability distributions in statistics?

<p>They help in estimating the likelihood of various outcomes. (D)</p> Signup and view all the answers

Which of the following statements can be classified as a hypothesis?

<p>Increased ice cream sales correlate with summertime. (D)</p> Signup and view all the answers

Which statement uses a predictor variable correctly in a hypothesis?

<p>Exercise improves health outcomes. (D)</p> Signup and view all the answers

What is a common mistake when formulating a hypothesis?

<p>Assuming a relationship without evidence. (D)</p> Signup and view all the answers

What is a two-tailed test primarily used for?

<p>Determining whether a relationship can be expected in either direction. (A)</p> Signup and view all the answers

What is true about the sampling distribution of the mean if the population is normally distributed?

<p>It is normally distributed with a mean equal to the population mean. (D)</p> Signup and view all the answers

What does the Central Limit Theorem (CLT) state about large sample sizes?

<p>The sampling distribution will be approximately normally distributed for sample sizes greater than 30. (C)</p> Signup and view all the answers

How is the standard error of the sampling distribution calculated?

<p>It equals the standard deviation of the sample divided by the sample size. (A)</p> Signup and view all the answers

What is meant by the confidence level in parameter estimation?

<p>It represents the frequency of all estimations expected to capture the true parameter. (D)</p> Signup and view all the answers

If a confidence level is set to 95%, what is the corresponding risk level (α)?

<p>0.05 (A)</p> Signup and view all the answers

Which of the following best describes the relationship between confidence level and significance level (α)?

<p>Confidence level is equal to 1 minus the significance level. (A)</p> Signup and view all the answers

What do critical values in statistical analysis indicate?

<p>They define a boundary for acceptance of the null hypothesis. (B)</p> Signup and view all the answers

What does a higher confidence level imply regarding the likelihood of wrong estimations?

<p>It decreases the likelihood of wrong estimations. (A)</p> Signup and view all the answers

What is the primary difference between alternative hypothesis (H1) and null hypothesis (H0)?

<p>H1 predicts an effect, while H0 predicts no effect. (A)</p> Signup and view all the answers

What does rejecting the null hypothesis (H0) imply?

<p>There is evidence against H0. (A)</p> Signup and view all the answers

In NHST (Null Hypothesis Significance Testing), what is assessed?

<p>The probability of observing the sample data assuming H0 is true. (D)</p> Signup and view all the answers

Which of the following is true regarding the relationship between H1 and H0?

<p>H0 nullifies the predictions made by H1. (B)</p> Signup and view all the answers

What outcome does failing to reject the null hypothesis (H0) suggest?

<p>There is insufficient evidence to support H1. (B)</p> Signup and view all the answers

Which of the following statements best describes statistical hypothesis testing?

<p>It compares the likelihood of observed data under H0 and H1. (D)</p> Signup and view all the answers

What is the implication of collecting evidence against H0?

<p>H1 remains a possibility. (A)</p> Signup and view all the answers

Which of the following pairs accurately defines the hypotheses in the given example?

<p>H1: Heavy metal fans have above average IQ; H0: Heavy metal fans do not have above average IQ. (C)</p> Signup and view all the answers

Flashcards

Probability (frequency) distribution

A function that describes the likelihood of different values of a random variable.

Statistical inference

The process of analyzing sample data to make inferences about the population.

Population parameters

Specific characteristics of a population, like average age or percentage of female customers.

Parameter estimation

Using sample data to estimate population parameters.

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Hypothesis

A statement about a population parameter that we want to test.

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Hypothesis formulation

Formulating a hypothesis about a population parameter.

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Statistical model

A statistical model using a test statistic to test hypothesis and make inferences about the population.

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Test statistic

A value used to summarize sample data and test hypothesis.

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Probability Distribution

A type of probability distribution where the probabilities of events are represented by a continuous curve.

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Normal Distribution

A specific probability distribution with a bell-shaped curve. It's symmetrical, with the mean, median, and mode all at the center.

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Standard Normal Distribution

A special case of the Normal Distribution where the mean is 0 and the standard deviation is 1. This allows easy comparison across different normal distributions.

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68-95-99 Rule

A rule used to quickly estimate probabilities in a normal distribution based on its standard deviation. It states that roughly 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations of the mean, respectively.

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Standardizing a Normal Distribution

The process of transforming a normal distribution with any mean and standard deviation into the standard normal distribution.

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Mean

The average value in a data set. In a normal distribution, the mean is at the center of the bell curve.

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Standard Deviation

A measure of how spread out the data is in a distribution. In a normal distribution, the standard deviation determines the width of the bell curve.

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Median

The value that divides the dataset in half. In a normal distribution, the mean, median, and mode are all equal.

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Sampling distribution of the mean

The distribution of sample means calculated from repeated samples drawn from a population. It describes the likelihood of obtaining different sample means.

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Standard error (of the mean)

The standard deviation of the sampling distribution of the mean. It measures the variability of sample means around the population mean.

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Central Limit Theorem (CLT)

A statistical concept that states that the sampling distribution of the mean will be approximately normal for large sample sizes (n>30), regardless of the population distribution. This is due to the averaging effect in repeated sampling.

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Confidence level

The level of confidence we have that our estimated parameter value includes the true population parameter. This is the probability of obtaining a sample that leads to an estimate containing the true value.

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Significance level (α)

The probability of making a wrong decision when testing a hypothesis. It's the chance that our sample leads us to reject the null hypothesis when it is actually true.

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Type I error

A type of error that occurs when we reject the null hypothesis when it is actually true.

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Critical values

The values on a probability distribution that correspond to the chosen confidence level or significance level. They define the boundaries of the critical region in hypothesis testing.

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Confidence interval

A range of values that we are confident contains the true population parameter based on the sample data and the chosen confidence level.

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Directional hypothesis

A hypothesis stating a specific direction of effect. It predicts not only whether there's an effect, but also if it's positive or negative. Example: 'People who exercise regularly have higher levels of happiness' (positive direction).

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Non-directional hypothesis

A hypothesis that simply predicts an effect between variables, but doesn't specify the direction of the effect. Example: 'There is a relationship between exercise and happiness' (no direction specified)

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Standard Deviation (SD)

A statistical measure of how spread out a set of data is from the mean. It tells us how much variation there is within the data

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Sample

A subset of a population that is used to collect data and make inferences about the whole population.

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Population

The entire group of individuals or items that we are interested in studying. All the things we want to analyze.

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Sampling Error

The difference between a sample statistic and the corresponding population parameter. This discrepancy often occurs because we don't have data on every individual in the population.

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Sample Statistic

A statistical measure based on data from a sample, used to estimate the corresponding population parameter.

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Null hypothesis (H0)

States that there is no effect or difference in the population. It is the opposite of the alternative hypothesis.

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Alternative hypothesis (H1)

States that there is an effect or difference in the population. This is what we typically want to support with our analysis.

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Rejecting or failing to reject the null hypothesis

A statistical test aims to gather evidence to either reject or fail to reject the null hypothesis. It never directly proves the alternative hypothesis.

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p-value

The probability of observing our sample data, assuming the null hypothesis is true. This helps determine the likelihood of the data if the null is actually correct.

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NHST (Null Hypothesis Significance Testing)

A common approach used in hypothesis testing. It involves analyzing sample data and calculating a test statistic to assess whether the null hypothesis is likely to be true.

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Study Notes

Data Analysis for Marketing Decisions

  • Session 2: Statistical Inference I Focuses on Parameter Estimation and Hypothesis Formulation.
  • Statistical Inference: Analyzing sample data to make inferences about a larger population.
  • Steps in Statistical Inference:
    • Identify the specific population characteristics (parameters)
    • Gather contrasts, comparisons, and associations
    • Derive estimates from the sample
    • Test hypotheses about those estimates
    • Fit appropriate statistical models using a test statistic
    • Analyze the sample data and the test statistics, considering the probability distributions to make inferences about the population.

Probability Distribution

  • Probability Distribution: A function describing the likelihood of different outcomes for a random variable. It's based on the underlying probability distribution.
  • Example Distribution: This presentation shows a frequency distribution of COVID-19 infection counts, split between individuals with and without masks.
  • Normal Distribution: A common distribution, characterized by specific properties (a symmetrical curve & 68-95-99.7 empirical rule). Means, medians, and modes are the same (symmetric) for this distribution

Normal and Standard Normal Distribution

  • Normal Distributions: All normal distributions share common properties defined by the 68-95-99.7 empirical rule.
  • Standard Normal Distribution: A normal distribution where the mean is zero and the standard deviation is one. This allows standardising any variable to make comparison easier.
  • Example of Application: Determining the likelihood of a student scoring below a certain mark based on known mean and standard deviation for a class.
  • Standardization (Z-scores): - Used to convert any normal distribution to the standard normal distribution when analyzing populations. - Formula: z= (x-μ)/σ. Where z is the z-score, x is the observed value, μ is the mean, and σ is the standard deviation

Example Application

  • Student Scores: An example illustrates how to calculate the probability a student scores below 58 given a mean and standard deviation.
  • Calculating Probability: Calculate likelihood using standardized values & a z-table.
  • Confidence Level:
  • 72.57% of students are expected to score below.
  • 27.43% of student scores are above.

Statistical Inference

  • Wait a Minute! Operating on a sample may not represent the entire population
  • Sampling Error: Implies error associated with selecting samples. This error can be calculated and considered using confidence level.
  • Population Parameter vs. Sample Statistic: - Population parameter: The value you are trying to estimate about the entire population, unknown value. - Sample statistic: The value calculated from the sample data, known and known.

Parameter Estimation

  • Collecting Data: This involves collecting data to find sample statistics relevant to population parameters to estimate the population parameter.
  • Mean, Proportions, etc: Estimate population parameters using statistics found using the sample. Example parameters include sample mean (X̄).

Sampling Distribution & Standard Error

  • Sampling Distribution: Probability distribution of a given sample statistic (e.g., the mean).
  • Mean of Sampling Distribution: Equal to the true population mean.
  • Standard Error (SE): This is the standard deviation of a sampling distribution.
  • Approximation: Can be approximated using a standard deviation if large sample sizes are used.

Parameter Estimation - Confidence Level

  • Confidence Level: Frequency of getting estimations that include the true population parameter,
  • Risk Level a (alpha): Likelihood the estimation is incorrect, the opposite of the confidence level.
  • Significance Level: A level of risk to incorrect estimations accepted (e.g., 1%).
  • Critical Values (Z-Scores): Points on the probability distribution that define the confidence interval.

Parameter Estimation - Confidence Interval

  • Confidence Interval: Range of values that, with a given confidence (e.g., 95%), likely contains the true population parameter.
  • Formula: μ = X̄ ± Z α/2 * S/ √n. where X̄ is the sample mean, S is the standard sample deviation, n is the sample size, and Z α/2 is the critical value corresponding to your chosen significance level (e.g., 95%). This formula uses the standard error.

Hypothesis Formulation

  • Hypothesis: A statement about the relationship between two or more variables. It can be empirically tested from data.
  • Types of Hypotheses:
    • Directional: Indicates the expected direction of the relationship (either positive or negative).
    • Non-directional: Does not specify the direction of the relationship.
  • Null Hypothesis (Ho): Opposite statement to the research hypothesis in a study. It predicts that no relationship or effect exists.

Types of Hypotheses (Detailed)

  • Alternative Hypothesis (H₁): A predictive statement; Often that there is a relationship between 2 or more variables.
  • Null Hypothesis (H₀): States there is no relationship or effect.

Types of Hypothesis (Details: testing)

  • Testing Hypotheses:
  • Never prove alternative: Statistical evidence is used to provide evidence against the null hypothesis.
  • Rejecting Ho: Doesn't prove H₁ (it merely maintains it)
  • NHST (Null Hypothesis Significance Testing): NHST involves assessing the likelihood of the data, given that the null hypothesis is true.
  • p-value: Probability of getting the results you got, or something more extreme, when the null hypothesis is true.
  • High p-value: Doesn't provide evidence against a null hypothesis. Low p-value provides good evidence.

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Description

This quiz covers key concepts in statistical inference, focusing on parameter estimation and hypothesis formulation. You will learn how to analyze sample data and derive estimates to make inferences about larger populations. Additionally, it emphasizes fitting statistical models and understanding probability distributions.

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