Midterm Exam Prep

• The midterm exam will be available on Canvas on Thursday, March 27th, at 4:30 pm EST.
• Students have 90 minutes to complete the exam within a 24-hour period.
• The exam closes on Friday, March 28th, at 4:30 pm EST.
• Notes and calculators are permitted during the exam.
• The midterm is cumulative, covering all topics from the beginning of the semester.
• Reviewing class meeting highlights and re-attempting homework and quiz problems are recommended for preparation.
• Class slides should be reviewed, focusing on sections needing quick access during the test.
• All formulas and tables required for the test will be provided.
• Access the test via the "Quizzes" tab on Canvas, selecting "Midterm Exam".

Key Topics for the Midterm:

• Alpha level: The probability of rejecting the null hypothesis when it is true, also known as a Type I error.
• Central Limit Theorem: States that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution.
• Confidence Interval: A range of values likely to contain a population parameter with a certain level of confidence.
• Confidence Level: The probability that a confidence interval contains the true population parameter.
• Criteria for calculating test statistics for means and proportions: Conditions that must be met to ensure the validity of the test statistic.
• Critical value: A point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis.
• Cumulative Density Function: A function giving the probability that a random variable is less than or equal to a certain value.
• Data types and levels of measurements: Includes nominal, ordinal, interval, and ratio scales, and qualitative vs. quantitative data.
• Descriptive statistics: Methods for summarizing and describing the main features of a data set.
• Frequency Distribution: Shows the number of occurrences of each value in a data set.
• Frequency Table: A table that lists each category of data and the number of occurrences for each category.
• Hypothesis Testing: A method for testing a claim or hypothesis about a population parameter using sample data.
• Hypothesized mean or proportion: The value assumed for the population mean or proportion in the null hypothesis.
• Inferential statistics: Methods for drawing conclusions about a population based on sample data.
• Likelihood of test statistics: The probability of obtaining a test statistic as extreme as, or more extreme than, the one actually observed, assuming the null hypothesis is true.
• Margins of Error: The range of values above and below the sample statistic in a confidence interval.
• Mean of sample means: The average of the means from multiple samples, which should approximate the population mean according to the Central Limit Theorem.
• Measures of centers (mode, median, mean) as represented on different graphs: Different measures of central tendency and their representation in graphs.
• Normal Probability Distribution: A symmetric, bell-shaped distribution with specific properties.
• Null and Alternate hypotheses: The null hypothesis is a statement of no effect or no difference, while the alternate hypothesis is a statement that contradicts the null hypothesis.
• P-value: The probability of obtaining a test statistic as extreme as, or more extreme than, the one actually observed, assuming the null hypothesis is true.
• Point estimate: A single value that is used to estimate a population parameter.
• Population mean: The average value of a variable in the entire population.
• Population parameter: A numerical value that describes a characteristic of a population.
• Population standard deviation: A measure of the spread or variability of data in the entire population.
• Probability (Experimental and Theoretical): Experimental probability is based on actual experiments, while theoretical probability is based on mathematical calculations.
• Probability Distribution Function: A function that describes the probability of a continuous random variable falling within a particular range of values.
• Probability Mass Function: A function that gives the probability that a discrete random variable is exactly equal to some value.
• Probability of random variables: The likelihood of different outcomes for a random variable.
• Random variable: A variable whose value is a numerical outcome of a random phenomenon.
• Range: The difference between the largest and smallest values in a data set.
• Rejection Region: The set of values for the test statistic that leads to rejection of the null hypothesis.
• Relative frequency histogram: Displays the proportion of occurrences for each class interval in a data set.
• Sample mean: The average value of a variable in a sample.
• Sample standard deviation: A measure of the spread or variability of data in a sample.
• Sample statistic: A numerical value that describes a characteristic of a sample.
• Sampling Distribution: The distribution of a statistic (e.g., sample mean) from multiple samples of the same size taken from the same population.
• Significance level: The probability of rejecting the null hypothesis when it is true (Type I error).
• Skewness: A measure of the asymmetry of a distribution.
• Standard deviation: A measure of the spread or variability of data around the mean.
• Standard Error of sample means: The standard deviation of the sampling distribution of sample means.
• Standard Normal Probability Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
• T-distribution: A probability distribution that is used to estimate population parameters when the sample size is small or when the population standard deviation is unknown.
• T-scores: A type of standard score that tells how many standard deviations away from the mean a particular score is.
• Variance: A measure of the spread or variability of data; the square of the standard deviation.
• z-scores: A measure of how many standard deviations an element is from the mean.

Assignment 1

• Question 1 asks to categorize the data types and measurement levels of statements.

  • Statement A: Anxiety scale score of 16 is twice as anxious as a score of 8.
    • Data type: Quantitative or Numerical
    • Measurement level: Ratio
  • Statement B: Participants identify as omnivore, vegetarian, vegan, or fruitarian.
    • Data type: Qualitative or Categorical
    • Measurement level: Nominal
  • Statement C: The difference between 6 and 8 is equivalent to the difference between 13 and 15.
    • Data type: Quantitative or Numerical
    • Measurement level: Interval
  • Statement D: Students earn a failing, passing, or distinction grade.
    • Data type: Qualitative or Categorical
    • Measurement level: Ordinal

• Question 2 involves a graphical representation of midterm scores.

  • The dataset has four modes around 50, 62, 73, and 78.
  • The median falls around the 13th data point which is approximately 72.
  • The mean should be close to the median because the dataset is roughly symmetrical with most data points in the middle.

• Question 3 deals with descriptive measures on a 20-point quiz.

  • Correcting an 18 to 16 changes the mode from 18 to no mode.
  • The median remains between the 5th and 6th data point, so it stays at 12.
  • The mean changes as it considers the value of every data point.
  • The range does not change as the maximum and minimum scores are the same.
  • Both variance and standard deviation will likely change because the mean changes.

• Question 4 addresses the research process.

  • It starts with an observation of interest.
  • Ensuring no bias requires the collection of data.
  • A researchable question is formulated based on the unbiased observation.
  • A theory is provided to explain the observation.
  • A testable hypothesis is made based on the theory.
  • The type of data (Qualitative or Quantitative) to collect is determined.
  • A decision on how to collect the data needs to be made.
  • A brief description of how to analyze the data is provided.
  • The analysis of the data should inform your theory.

• Students were asked to download R and RStudio.

Discussion: Harking, Sharking, and Tharking

• Hollenbeck and Wright argue that "Tharking can promote the effectiveness and efficiency of both scientific inquiry and cumulative knowledge creation" and claim that the practice of Tharking is "beneficial to scientific progress and, in many cases, ethically required".

Assignment 2 Responses

• Question 1 focuses on measures of center (mode, median, mean) as "typical" scores.
  • The mean is defined in terms of distances of data points from the center, where the total distance from the mean to points below is equal to the total distance to points above.
  • The mean uses all dataset measurements in its calculation.
  • The mode is always included in the dataset.
• Question 2 requires indicating which measure of center can be used with a specific data type.
  • Nominal data type can use Mode: Yes, Median: No, Mean: No.
  • Ordinal data type can use Mode: Yes, Median: Sometimes, Mean: No.
  • Discreet data type can use Mode: Yes, Median: Sometimes, Mean: Sometimes.
  • Continuous data type can use Mode: Yes, Median: Yes, Mean: Yes.
  • Interval data type can use Mode: Yes, Median: Yes, Mean: Yes.
  • Ratio data type can use Mode: Yes, Median: Yes, Mean: Yes.
• Question 3 compares the mode, median, and mean of different frequency distributions.
  • A normal frequency distribution has the mode, median, and mean close to each other.
  • Outliers in a right-skewed frequency distribution will pull the mean to the right.
  • Outliers in a left-skewed frequency distribution will pull the mean to the left.
• Question 4 relates frequency distributions to probability.
  • A person who attended the car show is Not very likely to be 70 years old because only 3 people ages 70 and above attended.
  • A 30-year-old is Very likely to have attended because most of the attendees are ages 30 to 34 years.
  • The probability that an attendee at the car show is aged 15 to 19 years is 10/172 = 0.058.
• Question 5 involves flipping three fair coins.
  • The sample space is Ω = {𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝑇𝐻𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝑇, 𝑇𝑇𝐻, 𝑇𝑇𝑇}.
  • The theoretical probability of getting at least two heads is 1/2 = 0.5.

Assignment 4 Responses

• Variables measured in the survey are gender, age, systolic blood pressure, and diastolic blood pressure.
  • Gender is qualitative and nominal.
  • Age is quantitative and ratio.
  • Systolic and diastolic blood pressure are quantitative and ratio.
  • The data is multivariate.
• Histograms of systolic blood pressure data for males and females were constructed.
  • More males are at risk for a stroke or a heart attack, given histograms. They have higher systolic blood pressure (>120) than women.
• Three Sampling Distributions were created. -Sample Size, n=5 - Mean: 114.782 - Standard deviation: ]5.962651
  • Sample Size, n=50
    • Mean : 114.4064
    • Standard deviation: 1.80838
  • Sample Size, n=100
    • Mean: 114.2009
    • Standard deviation: 1.339379
• The sampling distribution is used to calculate the confidence interval at the 98% confidence level.
  • The point estimate is chosen randomly from the 100 means of the samples of 50 data points (𝑥̅ = 115.38).
  • The Standard Error is 1.80838
  • The Critical Value is 2.33
• A concluding statement about the confidence interval in regard to the population mean (111.17 to 119.60).

Assignment 5 Responses

• Question 1: Identify the sample statistic.

  • The sample statistic used to create a confidence interval for the population parameters in situations.
  • Mean: Process is measured in minutes, so it's a continuous variable (manufacturing process).
  • Mean: Weight is measured in lbs, so it's a continuous variable (oat grain distributor).
  • Proportion: 16-liter bottles are countable (discreet) (bottling factory).

• Question 2: Criteria for the Use of Scores :Confidence interval

  • Calculations - Population Mean
    • The sampling distribution is normal
    • SE, 𝜎!̅ 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 We may consider using 𝑠!̅.
    • Using 𝑠!̅ to estimate 𝜎!̅ adds uncertainty, so we use a different critical value, a 𝑡 − 𝑠𝑐𝑜𝑟𝑒, 𝑡 ∗ from a t-distribution table.
  • Calculations - Population Proportion
    • The sampling distribution is normal.𝑝̂ =
    • 𝑛𝑝̂ ≥ 10; # of successes ≥ 10
    • 𝑛𝑞; ≥ 10; # of failures ≥ 10
    • (4) 2) SE 𝜎$% is known We may consider using 𝑠!̅.Since 𝜎 is unknown, use & √( as 𝑠 is an unbiased estimator

• Question 3 - Calculating the Point Estimate

  • Cumulative Probability
    • 𝑝̂ = 0.68
    • 𝑧 ∗ B𝐶𝐿 + )

• F = 𝑧 ∗ B0.95 + +.+-

• F = 𝑧 ∗ (. 95 +.025) = 𝑧 ∗ (0.975) = 1.96

  • What is the standard error
    • Two criteria for using 𝑧 − 𝑠𝑐𝑜𝑟𝑒𝑠 for population proportions

    • (no points lost on assignment for not checking assumptions)
      

    • The sampling distribution is normal.𝑝̂ =
    • 𝑛𝑝̂ ≥ 10; # of successes ≥ 10

    •   𝑛𝑞; ≥ 10; # of failures ≥ 10
      

    • (7) We know 𝜎$%
    • 𝜎$% = 𝜎√𝑛 → 𝑠√𝑛 = =𝑝̂ ∙ 𝑞; 𝑛
    • Since 𝜎 is unknown, use & √( as 𝑠 is an unbiased estimator

  -At the 95% confidence level, the Confidence Interval is likely to contain the true proportion of all students who take Intermediate Statistics to fulfill a degree program requirement.

• Question 4 -Confidence interval , Calculating the Point Estimate

  •      Calculate
    

    • The two criteria to use a z-score are met

    •        (no points lost on assignment for not checking assumptions).
      

    •   The sampling distribution is normal.
      

    •             (i)  𝑛𝑝̂ ≥ (1011)(0.64) = 647 ≥ 10; # of successes ≥ 10
      

    •                     𝑛𝑞; ≥ (1011)(0.36) = 364 ≥ 10; # of successes ≥ 10
      

    •         (9)   We know 𝜎$%
      

    •   𝜎$% = 𝜎√𝑛 → 𝑠√𝑛 = =𝑝̂ ∙ 𝑞; 𝑛
      

    • Since 𝜎 is unknown, use & √( as 𝑠 is an unbiased estimator

    •          The true proportion of voters who are likely to vote for the independent candidate.
      

• Question 5 - Calculating the Point Estimate - .The sampling distribution will be normally distributed for a - Sample size of 40 (≥ 30); however, the SE, 𝜎!̅ is 𝑢𝑛𝑘𝑛𝑜𝑤𝑛. The use of 𝑠!̅ to estimate 𝜎!̅ adds more uncertainty, so we use a 𝑡 ∗ (from a t-distribution table). - = 0.145 - We know 𝜎$%

• At the 80% confidence level, the Confidence Interval 𝟎. 𝟏𝟒𝟒 𝒕𝒐 𝟎. 𝟏𝟒𝟔 moles is likely to contain the mean amount of copper precipitated from the solution over a half-hour*