Reference:

https://www.r-bloggers.com/2022/05/one-sample-proportion-test-in-r-complete-guide/

https://zhuanlan.zhihu.com/p/349916370

One sample proportion test in R, when there are just two categories, the one proportion Z-test is used to compare an observed proportion to a theoretical one.

have a population that is half male and half female (p = 0.5 = 50%). Some of these total (n = 160), including 100 males and 60 females, acquired a spontaneous malignancy.

[Malignancy (from Latin male ‘badly’, and -gnus ‘born’) is the tendency of a medical condition to become progressively worse.]

Question: We’d like to know if cancer affects more men than women.

The observed male proportion (po) is 100/160. The observed female percentage (q) is 1-po. The predicted male proportion (pe) is 0.5. (50 percent ) A total of 160 observations (n) were made.

The following are the relevant alternative hypothesis (H1)

H1:po≠pe (different); hypotheses 1 H1:po>pe (greater); hypotheses 2 H1:po<pe (less); hypotheses 3 Note that:

Two-tailed tests are used to test hypotheses 1. One-tailed tests are used to test hypotheses 2 and 3.

To do a one-proportion test, use the R methods binom.test() and prop.test():

Calculate the exact binomial test with binom.test(). When the sample size is small, prop.test() is recommended.

binom.test(x, n, p = 0.5, alternative = "two.sided")
prop.test(x, n, p = NULL, alternative = "two.sided", correct = TRUE)

x: the number of successes n: the total number of trials p: the probability to test against. correct: a logical indicator of whether Yates’ continuity correction should be used if at all practicable.

prop <- prop.test(x = 100, n = 160, p = 0.5, correct = FALSE)
prop

## 
##  1-sample proportions test without continuity correction
## 
## data:  100 out of 160, null probability 0.5
## X-squared = 10, df = 1, p-value = 0.001565
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.5478817 0.6962568
## sample estimates:
##     p 
## 0.625

If you wish to see if the percentage of men with cancer is less than 0.5 (one-tailed test), enter:

prop.test(x = 100, n = 160, p = 0.5, correct = FALSE, alternative = "less")

## 
##  1-sample proportions test without continuity correction
## 
## data:  100 out of 160, null probability 0.5
## X-squared = 10, df = 1, p-value = 0.9992
## alternative hypothesis: true p is less than 0.5
## 95 percent confidence interval:
##  0.0000000 0.6853844
## sample estimates:
##     p 
## 0.625