The built-in R dataset ‘swiss’ gives fertility measures and socio-economic indicators for 47 French-speaking provinces of Switzerland for the period around 1888. The country was undergoing a demographic transition and fertility levels were starting to fall from the typically high levels characteristic of underdeveloped countries. The Fertility column gives a common standardized fertility measure, while the other variables give proportions of the population.

library(psych)
describe(swiss)
##                  vars  n  mean    sd median trimmed   mad   min   max
## Fertility           1 47 70.14 12.49  70.40   70.66 10.23 35.00  92.5
## Agriculture         2 47 50.66 22.71  54.10   51.16 23.87  1.20  89.7
## Examination         3 47 16.49  7.98  16.00   16.08  7.41  3.00  37.0
## Education           4 47 10.98  9.62   8.00    9.38  5.93  1.00  53.0
## Catholic            5 47 41.14 41.70  15.14   39.12 18.65  2.15 100.0
## Infant.Mortality    6 47 19.94  2.91  20.00   19.98  2.82 10.80  26.6
##                  range  skew kurtosis   se
## Fertility        57.50 -0.46     0.26 1.82
## Agriculture      88.50 -0.32    -0.89 3.31
## Examination      34.00  0.45    -0.14 1.16
## Education        52.00  2.27     6.14 1.40
## Catholic         97.85  0.48    -1.67 6.08
## Infant.Mortality 15.80 -0.33     0.78 0.42

“Fertility” has a mean of 70.14 and a standard deviation of 12.49.

pairs(swiss)

We will use a confidence level of 95% to find the confidence interval for Fertility and the other variables. Since n>30, a z test will be the appropriate to use for this dataset.

The significance level alpha = 0.05.

FERTILITY: COnfidence level upper:

#Z test: xbar+qnorm(0.95)*standard error  #standard error=standard dev/sqrt(n)

#xbar=70.14, s.d.= 12.49
70.14+qnorm(0.95)*(12.49/sqrt(47))
## [1] 73.13668

Confidence interval lower: #xbar+qnorm(0.05)*standard error

70.14+qnorm(0.05)*(12.49/sqrt(47))
## [1] 67.14332

Our level of certainty about the true mean for the vaariable FERTILITY is 95% in predicting that the true mean is within the interval between 67.143 and 73.137.

Let’s look at the confidence intervals for the true mean for the remaining variables:

AGRICULTURE:

COnfidence level upper:

#xbar = 50.66, s.d.=22.71

50.66+qnorm(0.95)*(22.71/sqrt(47))
## [1] 56.10873

Confidence interval lower:

50.66+qnorm(0.05)*(22.71/sqrt(47))
## [1] 45.21127

EXAMINATION:

COnfidence level upper:

#xbar = 16.49, s.d.=    7.98    

16.49+qnorm(0.95)*(7.98/sqrt(47))
## [1] 18.40461

Confidence interval lower:

16.49+qnorm(0.05)*(7.98/sqrt(47))
## [1] 14.57539

EDUCATION:

COnfidence level upper:

#xbar = 10.98, s.d.=9.62

10.98+qnorm(0.95)*(9.62/sqrt(47))
## [1] 13.28809

Confidence interval lower:

10.98+qnorm(0.05)*(9.62/sqrt(47))
## [1] 8.671906

CATHOLIC:

COnfidence level upper:

#xbar = 41.14, s.d.=    41.70

41.14+qnorm(0.95)*(41.7/sqrt(47))
## [1] 51.14494

Confidence interval lower:

41.14+qnorm(0.05)*(41.7/sqrt(47))
## [1] 31.13506

INFANT MORTALITY:

COnfidence level upper:

#xbar = 19.94, s.d.= 2.91

19.94+qnorm(0.95)*(2.91/sqrt(47))
## [1] 20.63819

Confidence interval lower:

19.94+qnorm(0.05)*(2.91/sqrt(47))
## [1] 19.24181