4.1 The Theory

A confidence interval provides additional information on variability about the point estimate. A range of values for a population parameter are calculated using a single sample.

A specific confidence interval will either contain or will not contain the population parameter.

We often use a 95% level of confidence (LOC): 95% of the time, the population parameter should lie within the confidence interval constructed around the sample mean…assuming that any difference between the sample estimate and the population parameter is attributable to random sampling error alone.

For example, our estimate of unemployment among Bangladeshi is 5.9% (23/391 x 100). This estimate has an associated 95% confidence interval of 3.6% to 8.2%. In other words, we are 95% sure that the % unemployed Bangladeshis in the population as a whole falls in the range 3.6% – 8.2%.

Level of significance: \[ \alpha = {100\% - LOC} \]

So if 95%:

\[ 5\% = 100\% - 95\% \]

4.2 CI Calculation

\[95\% \:CI = \pm 1.96 \times se(p)\] where se(p) is the standard error of the sample estimate.

The formula for calculating the standard error for the sample estimate of a percentage is: \[se(p) = \sqrt{ \frac {p \times (100-p)} {N} } \] where p = the estimated percentage ( numerator / denominator x 100) and N = the size of the denominator upon which the percentage is based.

For Bangladeshis who are unemployed:

# recall the percentage
p <- prc_df[5,4]
p

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1    5.88

# denominator
NG <- qlfs %>%
      filter(!is.na(unemp), !is.na(EthnicGroup)) %>%
      count(EthnicGroup)
N <- NG[5,2]

# standard error    
se <- ( (p * (100 - p)) / N)^0.5
se

##    percent
## 1 1.189933

ci95_lb <- p - (1.96 * se)
ci95_ub <- p + (1.96 * se)
ci95_lb 

##    percent
## 1 3.550083

ci95_ub

##    percent
## 1 8.214623

4.3 Comparing CIs

Calculating the 95% CI for one percentage is all well and good, but often we want to calculate and compare the 95% CIs for multiple percentages at the same time:

Fig.2 CIs

p <- prc_df[,4]
se <- ((p * (100 - p)) / NG[,2])^0.5
ci95_lb <- p - (1.96 * se)
ci95_ub <- p + (1.96 * se)

prc_df <- data.frame(prc_df, ci95_lb, ci95_ub)
colnames(prc_df) <- c("unemp", "EthnicGroup", "n", "percent", "ci95_lb", "ci95_ub")

# plot
ggplot(data = prc_df, aes(y = EthnicGroup, x = percent)) +
     geom_point(stat = "identity") +
     geom_segment( aes( y = EthnicGroup, yend = EthnicGroup, x = ci95_lb, xend = ci95_ub) ) +
     # Add labels
    labs(title= paste(" "), x = "% of Unemployed", y = "Ethnic Group") +
    theme_classic() + 
      # Axis label size
    theme(axis.text=element_text(size=14))

We can draw various conclusions from this graph:

1. The narrower the confidence interval, the more confident we are about a given survey estimate.

2. We are more confident about the survey estimates of the unemployment rate for some ethnic groups than for others.

3. This is mainly a reflection of the size of the ethnic group sample. The fewer respondents from a given ethnic group in the survey, the wider the confidence interval.

4. If two confidence intervals overlap, then we can’t be 95% confident which unemployment rate is larger in the population as a whole eg. the survey estimate of the unemployment rate for Indians looks notably higher than the rate for the Chinese. However, since the confidence intervals overlap, we can’t be 95% confident that this is true in the population as a whole.

5. If two confidence intervals do not overlap, then we can be at least 95% confident that the one rate is higher than the other in the population as a whole eg. the 95% CIs of the survey estimates for the Black and Chinese unemployment rates do not overlap; therefore we are at least 95% confident that in the population as a whole the Black unemployment rate is higher than the Chinese unemployment rate.

5.1 The Theory

1. State your hypothesis eg. group percentages are equal. Two hypotheses:

Null hypothesis: Claim or assumption to be tested eg. group percentages are equal

Alternative hypothesis: Alternative claim if evidence doesn’t support the null hypothesis eg. group percetanges are different, or smaller than, or greater than

2. Establish a decision rule based a LOC (or significance) eg. 95% (or 5%):

No significant at 5%, if \(|z|\) < 1.96 (or p-value \(>\) 5%)

Significant at 5%, if \(|z| \ge\) 1.96 (or p-value \(\le\) 5%)

p-value: is the probability of an observed (or more extreme) result assuming that the null hypothesis is true. The smaller the p-value, the higher the significance indicating that the hypothesis under consideration may not adequately explain the observation.

3. Compute the test statistic (z)

4. Interpretation / conclusion of the results

Fig. 2. Hypothesis testing: Rejection regions.

5.2 Implementation

Testing the difference in unemployment between Chinese (3.5%) and Other Asian background (5.4%).

# get n of unemployed people
n_une <- prc_df[6:7, 3]
# total population
n_pop <- as.numeric(unlist(NG[6:7, 2])) # unlist simplifies the structure of the data to produce a vector
prop.test(x = n_une, n = n_pop)

## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  n_une out of n_pop
## X-squared = 1.5542, df = 1, p-value = 0.2125
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.046256039  0.007727997
## sample estimates:
##     prop 1     prop 2 
## 0.03458213 0.05384615

The output:

• the value of Pearson’s chi-squared test statistic.
• a p-value
• a 95% confidence intervals
• an estimated probability of success ie. the proportion of unemployed people in the two groups

The p-value of the test is 0.0.2125, which is greater than the significance level \(\alpha\) = 0.05. We can conclude that the proportion of unemployed people between Chineses and Other Asian background are not significantly different.

This equivalent to the conclusion we would reach using CIs.