One-Sample Proportions

Introduction

• In the last module, we discussed describing data.

• In this module, we are discussing statistical inference on one sample.

  • Mean / median

  • Proportion

  • Variance / standard deviation

• In this lecture, we are focusing on one-sample proportions.

R Setup

library(tidyverse)
library(ssstats)

• Recall our Zootopia data,

zootopia <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/1-zootopia.csv")

Confidence Interval for \pi

• (1–\alpha)100% CI for a population proportion, \pi

\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

• \hat{p} is the sample proportion (the point estimate)

• z_{\alpha/2} comes from the standard normal distribution (the critical value)

• \hat{p}(1-\hat{p}) is the variance of the Bernoulli distribution and n is the sample size (\sqrt{\hat{p}(1-\hat{p})/n} is the standard error of the mean)

Hypothesis Test for \pi

• Hypotheses

• \pi is the population proportion,

• \pi_0 is the hypothesized value of \pi.

Hypothesis Test for \pi

• Test Statistic

z_0 = \frac{\hat{p} - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}

• where

  • \hat{p} is the sample proportion

  • \pi_0 is the hypothesized value of \pi

  • n is the sample size

Statistical Inference on One Proportion (R)

• We will use the one_proportion() function from library(ssstats) to perform statistical inference.

dataset_name |> one_proportion(outcome = variable_name,
                                event = event_name_or_number,
                                p = hypothesized_value,
                                alternative = "two" | "less" | "greater",
                                alpha = alpha_level)

• where

  • outcome is the variable name of the outcome variable

  • event is the name or number of the event of interest

  • p is the hypothesized value of the population proportion (default = 0.05)

  • alternative is the alternative hypothesis (default = “two”)

  • alpha is the significance level (default = 0.05)

Example 1

• ZPD’s official staffing policy states that 50% of officers are assigned to Patrol, with the rest working Desk duty. As Zootopia’s data analyst, you believe that Patrol seems to make up a lot more than 50% of the department these days. We will use the zootopia dataset to answer:

  • Has the true proportion of officers in Patrol increased from 50%?

• First, we need to describe the data.

  • We start with point and interval estimates.

• Then, we will perform a hypothesis test to answer the question.

Example 1

• Point and interval estimates:

zootopia |> one_proportion(outcome = department)

Error in `one_proportion()`:
! argument "event" is missing, with no default

• Oops! We need to make sure we specify the event of interest. Let’s see how it’s coded in the department variable.

zootopia |> n_pct(department)

Example 1

• Point and interval estimates:

zootopia |> one_proportion(outcome = department,
                            event = "Patrol")

One-sample proportion

p̂ = 0.6 (18/30)

95% CI for π: (0.4247, 0.7753)

Hypotheses:
H₀: π = 0.5
H₁: π ≠ 0.5
Test statistic: z = 1.1
p-value: 0.273
Conclusion: Fail to reject the null hypothesis (p = 0.273 ≥ α = 0.050)

• The estimated proportion of officers in Patrol is 60% (95% CI: 42.5%, 77.5%).

Example 1

• Then, we will perform a hypothesis test to answer the question.

  • Has the true proportion of officers in Patrol increased from 50%?

• How do we translate this into hypotheses?

  • H_0: \pi \le 0.5
  • H_1: \pi > 0.5

• For the one_proportion() function, we will set

  • p = 0.5
  • alternative = "greater".

Example 1

• Has the true proportion of officers in Patrol increased from 50%?

zootopia |> one_proportion(outcome = department,
                            event = "Patrol",
                            p = 0.5,
                            alternative = "greater",
                            alpha = 0.05)

One-sample proportion

p̂ = 0.6 (18/30)

95% CI for π: (0.4247, 0.7753)

Hypotheses:
H₀: π ≤ 0.5
H₁: π > 0.5
Test statistic: z = 1.1
p-value: 0.137
Conclusion: Fail to reject the null hypothesis (p = 0.137 ≥ α = 0.050)

Example 1

• Hypotheses

  • H_0: \pi \le 0.5
  • H_1: \pi > 0.5

• Test Statistic and p-Value

  • z_0 = 1.1
  • p = 0.137

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Fail to reject H_0. There is not sufficient evidence to suggest that the current proportion of officers in Patrol has increased from 50%.

Example 2

• Zootopia’s Housing Authority claims that 40% of officers live in Downtown. As data analyst, you believe Downtown might actually make up a smaller proportion of the roster than the Authority’s number suggests. We will use the zootopia dataset to answer:

  • Is the true proportion of officers living in Downtown actually less than 40%?

• First, we need to describe the data.

  • We start with point and interval estimates.

• Then, we will perform a hypothesis test to answer the question.

Example 2

• Looking at what is in district,

zootopia |> n_pct(district)

• We know that the event of interest is coded as Downtown.

Example 2

• Point and interval estimates:

zootopia |> one_proportion(outcome = district,
                            event = "Downtown")

One-sample proportion

p̂ = 0.2667 (8/30)

95% CI for π: (0.1084, 0.4249)

Hypotheses:
H₀: π = 0.5
H₁: π ≠ 0.5
Test statistic: z = -2.56
p-value: 0.011
Conclusion: Reject the null hypothesis (p = 0.011 < α = 0.050)

• The estimated proportion of officers living in Downtown is 26.7% (95% CI: 10.8%, 42.5%).

Example 2

• Then, we will perform a hypothesis test to answer the question.

  • Is the true proportion of officers living in Downtown actually less than 40%?

• How do we translate this into hypotheses?

  • H_0: \pi \ge 0.4
  • H_1: \pi < 0.4

• For the one_proportion() function, we will set

  • p = 0.4
  • alternative = "less".

Example 2

• Is the true proportion of officers living in Downtown actually less than 40%?

zootopia |> one_proportion(outcome = district,
                            event = "Downtown",
                            p = 0.4,
                            alternative = "less",
                            alpha = 0.05)

One-sample proportion

p̂ = 0.2667 (8/30)

95% CI for π: (0.1084, 0.4249)

Hypotheses:
H₀: π ≥ 0.4
H₁: π < 0.4
Test statistic: z = -1.49
p-value: 0.068
Conclusion: Fail to reject the null hypothesis (p = 0.068 ≥ α = 0.050)

Example 2

• Hypotheses

  • H_0: \pi \ge 0.4
  • H_1: \pi < 0.4

• Test Statistic and p-Value

  • z_0 = - 1.49
  • p = 0.068

• Rejection Region

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation

  • Fail to reject H_0. There is not sufficient evidence to suggest that the current proportion of officers living in Downtown is less than 40%.

Wrap Up

• In this lecture, we discussed statistical inference on one proportion.

• Fun fact: proportions are just a mean!

• In the next lecture, we will examine statistical inference on one variance or standard deviation.