Question 1

Answer the following two questions using the inference function. As always, write out the hypotheses for any tests you conduct and outline the status of the conditions for inference.

  1. Is there convincing evidence that Spain has seen a change in its atheism index between 2005 and 2012? Hint: Create a new data set for respondents from Spain. Form confidence intervals for the true proportion of athiests in both years, and determine whether they overlap.
spain_12 <- subset(atheism, nationality == "Spain" & year == 2012)
spain_05 <- subset(atheism, nationality == "Spain" & year == 2005)
inference(spain_12$response, est = "proportion", type = "ci", method = "theoretical", success = "atheist")
## Single proportion -- success: atheist 
## Summary statistics:

## p_hat = 0.09 ;  n = 1145 
## Check conditions: number of successes = 103 ; number of failures = 1042 
## Standard error = 0.0085 
## 95 % Confidence interval = ( 0.0734 , 0.1065 )
inference(spain_05$response, est = "proportion", type = "ci", method = "theoretical", success = "atheist")
## Single proportion -- success: atheist 
## Summary statistics:

## p_hat = 0.1003 ;  n = 1146 
## Check conditions: number of successes = 115 ; number of failures = 1031 
## Standard error = 0.0089 
## 95 % Confidence interval = ( 0.083 , 0.1177 )

The Confidence Intervals overlap. There doesn't seem to be a significant change in the Atheism Index in Spain from 2005 to 2012.

  1. Is there convincing evidence that the United States has seen a change in its atheism index between 2005 and 2012?
us_12 <- subset(atheism, nationality == "United States" & year == 2012)
us_05 <- subset(atheism, nationality == "United States" & year == 2005)
inference(us_12$response, est = "proportion", type = "ci", method = "theoretical", success = "atheist")
## Single proportion -- success: atheist 
## Summary statistics:

## p_hat = 0.0499 ;  n = 1002 
## Check conditions: number of successes = 50 ; number of failures = 952 
## Standard error = 0.0069 
## 95 % Confidence interval = ( 0.0364 , 0.0634 )
inference(us_05$response, est = "proportion", type = "ci", method = "theoretical", success = "atheist")
## Single proportion -- success: atheist 
## Summary statistics:

## p_hat = 0.01 ;  n = 1002 
## Check conditions: number of successes = 10 ; number of failures = 992 
## Standard error = 0.0031 
## 95 % Confidence interval = ( 0.0038 , 0.0161 )

The Confidence Intervals don't overlap. There seems to be a significant change in the Atheism Index in US from 2005 to 2012. 2012 seems to have more Atheists compared to that of 2005.

Question 2

If in fact there has been no change in the atheism index in the countries listed in Table 4, in how many of those countries would you expect to detect a change (at a significance level of 0.05) simply by chance?

39*0.05
## [1] 1.95

2 countries.

Question 3

Suppose you're hired by the local government to estimate the proportion of residents that attend a religious service on a weekly basis. According to the guidelines, the estimate must have a margin of error no greater than 1% with 95% confidence. You have no idea what to expect for \(p\). How many people would you have to sample to ensure that you are within the guidelines?

e = 0.01
sample.size.prop(e, P = 0.5, N = Inf, level = 0.95)
## 
## sample.size.prop object: Sample size for proportion estimate
## Without finite population correction: N=Inf, precision e=0.01 and expected proportion P=0.5
## 
## Sample size needed: 9604