SAT College

Ignaz Semmelweiss (1818-1865) was the doctor who first encouraged other doctors to wash their hands with disinfectant before touching patients. Before the new procedure was established, the rate of infection at Dr. Semmelweiss' hospital was about 10%. Afterward the rate dropped to about 1%. Assuming the population proportion of infections was 10%, find the probability that the sample proportion will be 1% or less, assuming a sample size of 200. Start by checking the conditions required for the Central Limit Theorem to apply.

Answers

Answer 1

The probability of getting a smaple proportion of 1 % or less is almost zero.

Hence, we can conclude that it is highly unlikely to observe a sample proportion of 1% or less assuming the population proportion is 10%.

How is this so ?

To check if the Central Limit Theorem   (CLT) applies, we need to verify this -

Random Sample: The patients should be randomly selected.

Sample Size -  The sample size is large enough (n≥30).

Independence - The sample should be less than 10% of the population.

Since the sample size is less than 10% of the population, the conditions for CLT are met.

To find th eprobability,

z = (P - p) / √(p x (1- p) / n )

where

P = 0.01 (sample proportion)

p = 0.10 (population proportion)

n = 200 (sample size)

z = ( 0.01 -  0.10 )  / √(0.10 x  0.90 /200)

z =  - 8.94

Learn more about probability:
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Answer 2
The conditions required for the Central Limit Theorem to apply are that the sample size must be large (at least 30), the data must be independent and randomly sampled from the population, and the population must be normally distributed. Since the sample size is 200, the data is independent and randomly sampled, and the population is assumed to be normally distributed, the Central Limit Theorem can be applied.

Using the Central Limit Theorem, the probability that the sample proportion will be 1% or less is 0.8413.

Related Questions

What is permissive mean

Answers

The term "permissive" generally means granting permission or allowing something to be done without objection or interference. It implies a degree of flexibility or leniency in terms of rules or regulations. For example, a permissive parent may allow their child to make their own decisions without strict rules or discipline. In contrast, an authoritarian parent may have strict rules and expect strict obedience from their child. In a legal context, a permissive law or regulation allows something to be done, but does not necessarily require it.

According to the Bureau of Labor Statistics, 71.9% of Young women enroll in college directly after high school graduation. Suppose a random sample of 200 female high school graduates is selected and the proportion who enroll in college is obtained.


a. What value should we expect for the sample proportion?

b. What is the standard error?

c. What effect would increasing the sample size to 500 have on the standard error?

Answers

a. The expected value for the sample proportion can be calculated as the population proportion: 71.9%, so we expect that approximately 71.9% of the 200 female high school graduates will enroll in college directly after high school graduation.

Expected value = population proportion * sample size = 0.719 * 200 = 143.8

Therefore, we expect approximately 144 female high school graduates to enroll in college directly after high school graduation.

b. The standard error of the sample proportion can be calculated as:

SE = sqrt[p(1-p)/n], where p is the population proportion and n is the sample size.

SE = sqrt[0.719(1-0.719)/200] = 0.033

Therefore, the standard error is 0.033.

c. The standard error is inversely proportional to the square root of the sample size. Increasing the sample size from 200 to 500 would reduce the standard error:

SE = sqrt[0.719(1-0.719)/500] = 0.023

Therefore, the standard error would decrease to 0.023 if the sample size increased to 500.
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