Activity 20

MATH 216: Statistical Thinking

Activity 1: Think-Pair-Share

$$T=\frac{\overline{x}-\mu }{s/\sqrt{n}}$$

1. Think individually about why the t-statistic is more appropriate for small samples compared to the z-statistic.
2. Pair with a neighbor to discuss their thoughts.
3. Share your insights with the class.

Activity 2: Quiz

• Question: Given a sample mean $\overline{x}=2.283$, sample standard deviation $s=0.950$, and sample size $n=6$, calculate the t-statistic for a population mean $\mu =2.0$.

Activity 3: Interactive Data Exploration

Prompt: Provide students with the following dataset and ask them to explore it using RStudio:

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blood_pressure <- c(1.7, 3.0, 0.8, 3.4, 2.7, 2.1)

_webr_editor_1 = Object {code: null, options: Object, indicator: it}

• Task: Calculate the sample mean $\overline{x}$ and sample standard deviation $s$.
• Visualization: Create a histogram of the data and overlay a normal distribution curve.

Activity 4: Group Activities with Real Data

• Dataset:

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printhead_failures <- c(1.14, 1.23, 1.32, 1.21, 1.25, 1.28, 1.31, 1.22,

1.26, 1.29, 1.33, 1.24, 1.27, 1.30, 1.34)

_webr_editor_2 = Object {code: null, options: Object, indicator: it}

• Task: Calculate the 99% confidence interval for the mean number of characters printed before the printhead fails.
• Interpretation: Discuss the implications of the confidence interval in the context of the problem.

Activity 5: Case Studies and Problem-Solving

• Scenario: A pharmaceutical company wants to estimate the mean increase in blood pressure for patients taking a new drug. They have a small sample of 6 patients with the following blood pressure increases (in points):

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blood_pressure_increases <- c(1.7, 3.0, 0.8, 3.4, 2.7, 2.1)

_webr_editor_3 = Object {code: null, options: Object, indicator: it}

• Task: Construct a 95% confidence interval for the mean increase in blood pressure. Discuss the assumptions required for this interval to be valid and whether they are reasonably satisfied.

Activity 6: Calculating Sample Size

Consider a large hospital that wants to estimate the average length of stay of its patients, $\mu$ . The hospital randomly samples $n=100$ of its patients and finds that the sample mean length of stay is $\overline{x}=4.5$ days. Also, suppose it is known that the standard deviation of the length of stay for all hospital patients is $\sigma =4$ days.

Initial 95% C.I.: $4.5\pm 0.78$ days; width $=1.56$ days.

Adjusting Sample Size for Precision

• Goal: Narrow C.I. width from 1.56 days to 0.50 day.
• Calculation leads to a required sample size of approximately ____ patients.

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# Enter your code here ...

_webr_editor_4 = Object {code: null, options: Object, indicator: it}

Downloading webR