A survey of 800 randomly selected adults in a country found that 656 of them believed that protecting the rights of those with unpopular views is a very important component of a strong democracy. Find and interpret a 95% confidence interval for the proportion of adults in the country who believe that protecting the rights of those with unpopular views is a very important component of a strong democracy.
Step 1: Summary
# From the question, write down sample size (n) and proportion (p hat)
n <- 800
p_hat <- 656/nStep 2: Checking the Central Limit Theorem
# The adults are "randomly selected", so we assume Independent & Random
# Thus, we only need to check Large Sample aka Succeses & Failures
# We do not have the population proportion (p), so use p hat!
# Expected No. Successes is AT LEAST 10: n * p >= 10
n * p_hat >= 10## [1] TRUE# Expected No. Failures is AT LEAST 10: n * (1 - p) >= 10
n * (1 - p_hat) >= 10## [1] TRUEStep 3: Construct the CI
# CI = [p hat - z crit * SE, p hat + z crit * SE]
# z crit is the critical value, which depends on the CONFIDENCE LEVEL
# For the 90% CI, the critical value is 1.645
# For the 95% CI, the critical value is 1.960
# For the 99% CI, the critical value is 2.576
z_crit95 <- 1.96
# For the one sample proportion case, the Standard Error (SE) is:
# SE = sqrt(p * (1 - p) / n)
# Again, we do not have p, so approximate with p hat instead:
# SE hat = sqrt(p hat * (1 - p hat) / n)
SE_hat <- sqrt(p_hat * (1 - p_hat) / n)
# Margin of Error (MoE): z_crit * SE
MoE95 <- z_crit95 * SE_hat
# Lower Bound: p hat - z_crit * SE = p hat - MoE
lower95 <- p_hat - MoE95
# Upper Bound: p hat + z_crit * SE = p hat + MoE
upper95 <- p_hat + MoE95
# Use c() to combine the bounds together
ci95 <- c(lower95, upper95)
ci95## [1] 0.7933772 0.8466228# The CI's width is the upper bound minus the lower bound
upper95 - lower95## [1] 0.05324566# The CI's width is also always equal to twice the MoE
2 * MoE95## [1] 0.05324566Answer: Delete this text and type in your answer.
Would a confidence interval with a confidence level of 90% be wider or narrower than one with a confidence level of 95%? Explain why or why not. Create both CIs using the previous question’s information to support your answer.
# With the approximated SE hat = sqrt(p hat * (1 - p hat) / n):
# CI = [p hat - z_crit * SE hat, p hat + z_crit * SE hat]
# We use the same sample. Thus, p hat, n, & SE hat remains unchanged!
# NOTE: DECREASING THE CONFIDENCE LEVEL DECREASES THE CRITICAL VALUE:
# For the 90% CI, the critical value is 1.645
# For the 95% CI, the critical value is 1.960
z_crit90 <- 1.645
# The steps to calculate the 90% CI is the same as above.
# The only exception is that we are using z crit = 1.645!
# Again, p hat and SE hat are the same since we use the same sample!
# Margin of Error (MoE): z_crit * SE
MoE90 <- z_crit90 * SE_hat
# Lower Bound: p hat - z_crit * SE = p hat - MoE
lower90 <- p_hat - MoE90
# Upper Bound: p hat + z_crit * SE = p hat + MoE
upper90 <- p_hat + MoE90
# Use c() to combine the bounds together
ci90 <- c(lower90, upper90)
ci90## [1] 0.7976558 0.8423442# Once again, we calculate the new CI's width in both ways
upper90 - lower90## [1] 0.044688332 * MoE90## [1] 0.04468833Answer: Delete this text and type in your answer.
Download and read births.csv. Sample the weights of 100
randomly selected births with replacement from the dataset. Based on the
sample, create and interpret a 95% confidence interval for the
proportion of births with a baby weight less than but not including 115
ounces.
Read in the births.csv data:
births_df <- read.csv("births.csv")
head(births_df)## Gender Premie weight Apgar1 Fage Mage Feduc Meduc TotPreg Visits Marital
## 1 Male No 124 8 31 25 13 14 1 13 Married
## 2 Female No 177 8 36 26 9 12 2 11 Unmarried
## 3 Male No 107 3 30 16 12 8 2 10 Unmarried
## 4 Female No 144 6 33 37 12 14 2 12 Unmarried
## 5 Male No 117 9 36 33 10 16 2 19 Married
## 6 Female No 98 4 31 29 14 16 3 20 Married
## Racemom Racedad Hispmom Hispdad Gained Habit MomPriorCond BirthDef
## 1 White White NotHisp NotHisp 40 NonSmoker None None
## 2 White White Mexican Mexican 20 NonSmoker None None
## 3 White Unknown Mexican Unknown 70 NonSmoker At Least One None
## 4 White White NotHisp NotHisp 50 NonSmoker None None
## 5 White Black NotHisp NotHisp 40 NonSmoker At Least One None
## 6 White White NotHisp NotHisp 21 NonSmoker None None
## DelivComp BirthComp
## 1 At Least One None
## 2 At Least One None
## 3 At Least One None
## 4 At Least One None
## 5 None None
## 6 None NoneSampling:
# IMPORTANT: DO NOT CHANGE THIS SEED NUMBER!
set.seed(12)
# From the weight column, randomly sample 100 values
sample_weights <- sample(x = births_df$weight, size = 100)
head(sample_weights)## [1] 91 147 113 121 77 142Construct the CI:
# Follow the steps above to construct the 95% CI with your sample!
# You can skip checking the conditions!
# You are given n = 100. Calculate the sample proportion p hat.
n <- 100
p_hat <- mean(sample_weights < 115)
# For the 95% CI, the critical value is 1.960
z_crit95 <- 1.96
# Approximate the Standard Error (SE) with p hat:
# SE hat = sqrt(p hat * (1 - p hat) / n)
SE_hat <- sqrt(p_hat * (1 - p_hat) / n)
# Margin of Error (MoE): z_crit * SE
MoE95 <- z_crit95 * SE_hat
# Lower Bound: p hat - z_crit * SE = p hat - MoE
lower95 <- p_hat - MoE95
# Upper Bound: p hat + z_crit * SE = p hat + MoE
upper95 <- p_hat + MoE95
# Use c() to combine the bounds together
ci95 <- c(lower95, upper95)
ci95## [1] 0.4423141 0.6376859Answer: Delete this text and type in your answer.