Statistics Practice Problems

I. Average Sleep for Pre-Med Majors

• Objective: Determine average sleep hours for pre-med majors at University of Texas.

• Survey Sample: Eight individuals.

• Data:

  • Hours of sleep: 6, 4, 7, 7, 6, 5, 7, 6

1. Calculations

• a. Mean Hours of Sleep:

  • Formula: $\text{Mean} (X) = \frac{\sum{X}}{n}$

  • Calculation: $(6 + 4 + 7 + 7 + 6 + 5 + 7 + 6) / 8 = 6$ hours

• b. Standard Deviation: Given as 1.07 hours.

• c. Standard Error of the Mean (SEM):

  • Formula: $\text{SEM} = \frac{\text{Standard Deviation}}{\sqrt{n}}$

  • Calculation: $\text{SEM} = \frac{1.07}{\sqrt{8}} \approx 0.38$

  • Confidence Interval (95%): Multiply SEM by 2:

  • $95\%\ CI = 2 \times 0.38 = 0.76$

• d. Bar Graph Representation:

  • Mean: 6 hours

  • Error Bars:

    • Upper Limit: $\text{mean} + 2 \text{SEM} = 6 + 0.76 = 6.76$

    • Lower Limit: $\text{mean} - 2 \text{SEM} = 6 - 0.76 = 5.24$

2. Harvard Study Comparison

• e. Prediction on Standard Deviation:

  • Prediction: It will be higher than University of Texas due to a wider spread of data: hours ranged from 2 to 12.

• f. Bonus: Calculate Harvard's standard deviation using Excel or calculator.

  • Given result: Standard deviation = 3.5

  • Comparison: Prediction matches; standard deviation is higher due to more variability in the data.

II. Water Consumption in Athletes

• Objective: Determine if football players consume more water than soccer players.

1. Hypothesis and Data Analysis

• a. Null Hypothesis (H0): There is no difference in the average ounces of water consumed between football and soccer players.

• b. Create a bar graph comparing sample means for both players, including 95% CI (± 2SEM).

• c. Conclusion:

  • Stronger evidence supports rejecting H0, as means differ and error bars are juxtaposed.

• Quantified Water Consumption:

  • Football: 70 ± 13 ounces

  • Soccer: 64 ± 10 ounces

III. Understanding Standard Error Across Different Sample Sizes

1. Data Set Comparison

• Given sample sizes:

  • Dataset 1: n = 15

  • Dataset 2: n = 30

  • Dataset 3: n = 45

a. Calculate Standard Error for Each Dataset

• Formula: $\text{Standard Error} = \frac{\text{Standard Deviation}}{\sqrt{n}}$

• Standard deviation for all datasets = 2.3

  • a. Dataset 1 (n=15): $\text{SE} = \frac{2.3}{\sqrt{15}} \approx 0.59$

  • b. Dataset 2 (n=30): $\text{SE} = \frac{2.3}{\sqrt{30}} \approx 0.42$

  • c. Dataset 3 (n=45): $\text{SE} = \frac{2.3}{\sqrt{45}} \approx 0.34$

2. Effect of Sample Size on Standard Error

• As sample size increases, standard error decreases.

  • Justification: Standard error decreases because the standard deviation is divided by the square root of sample size.

IV. Statistical Metrics for Various Data Sets

• Calculate Median, Variability and Box Plots

1. Identify Medians and Variability

• a. Median for Dataset A:

• b. Median for Dataset B:

• c. Determining which dataset has more variability:

2. Box and Whisker Plot Analysis

• Analyze student scores comparing medians, lowest, highest scores, and percentage distributions for specified thresholds:

  • a. Lowest score:

  • b. Highest score:

  • c. Median:

  • d. % of students scoring ≤ 70:

  • e. % of students scoring ≤ 80:

V. Melanoma Behavioral Statistics in Lizards

• Data Analysis of Lizard Push-Ups:

  • Sample: 11 lizards counted

  • Data Points: 300, 105, 400, 521, 411, 500, 550, 52, 315, 75, 370

  • Metrics: Minimum: , Q1: , Q2: , Q3: , Maximum: __

  • Definition of IQR:

VI. Lead Concentration in Rainwater Samples

1. Community Comparison

• Box and whisker plot exhibits lead concentrations from two communities.

• Determine lower median concentration community.

• Analyze variability.

  • a. Community A vs Community B for median and variability.

• Local health guideline awareness: is lead level of ≥ 10 µg/L included in either IQR?

VII. Social Media Habits of Students

• Researcher’s Hypothesis: 9th graders use more social media than 11th graders.

1. Data Comparison

• Sample size: 10 each from both grades.

  • a. 11th grade data points: 10, 25, 85, 37, 42, 44, 55, 56, 59, 201

  • b. 9th grade data points: 40, 75, 10, 54, 66, 90, 42, 205, 77, 84

• Construct box and whisker plots for both datasets.

• Comparison Outcomes: Median usage, IQRs for both grades.

• Predict which grade group utilizes more social media time on average: 9th graders 70.5 mins vs 11th graders 49.5 mins.

VIII. Cancer Research on HER2-postive Breast Cancer Using Drugs

1. Overview of Treatment Effectiveness

• Normal vs Cancer Cell Differences: Cancer cells possess more HER2 receptors than normal cells, facilitating rapid growth.

2. Processing Results

• a. Apoptosis following treatment with Herceptin and Perjeta; means and comparison results.

  • Herceptin Data: Mean apoptosis: 20.35, STDEV, SEM

  • Perjeta Data: Mean apoptosis: 25, STDEV, SEM

  • Interpretation of apoptosis data.

3. Combination Treatments Analysis

• Repeat procedure for Herceptin + Perjeta combining results, calculate means, 95% CI.

  • Comparison to validate treatment effectiveness against controls.

• Final Recommendation: Study combination treatment due to higher effectiveness with lower side effects than alternative treatments.