Question

According to Readers Digest, 49% of primary care doctors think their patients receive unnecessary medical care. Use z-table.

a. Suppose a sample of 290 primary care doctors was taken. Show the sampling distribution of the proportion of the doctors who think their patients receive unnecessary medical care.

E(p)=

O(p)=

b. What is the probability that the sample proportion will be within (plus/minus) 0.03 of the population proportion. Round your answer to four decimals.

c. what is the probability that the sample poportion will be within (plus/minus) 0.05 of the population proportion. Round your answer to four decimals.

Answer 1

The expected proportion (E(p)) of primary care doctors who think their patients receive unnecessary medical care is 0.49, and the standard error (O(p)) is 0.0294. The probability that the sample proportion will be within ±0.03 or ±0.05 of the population proportion can be found using the z-table after calculating the z-scores.

Step-by-step explanation:

The subject of this question is statistics, specifically the use of z-tables for determining probabilities regarding sampling distributions. In a sampling distribution showing the proportion of doctors who think their patients receive unnecessary medical care, we calculate the expected proportion (E(p)) and the standard error of the proportion (O(p)).

To find E(p) for the given situation, we use the provided population proportion of primary care doctors who think their patients receive unnecessary medical care:

• E(p) = population proportion = 0.49

To compute the standard error O(p), we use the formula:

O(p) = sqrt[p(1-p)/n]

Where p is the population proportion (0.49), and n is the sample size (290).

• O(p) = sqrt[0.49(1-0.49)/290]
• 
  `O(p) = sqrt[0.49 * 0.51 / 290]`
• O(p) = sqrt[0.2499 / 290]
• O(p) = sqrt[0.0008617]
• O(p) = 0.0294

To find the probability that the sample proportion is within ±0.03 of the population proportion, we determine the z-scores corresponding to p = 0.49 ± 0.03 using the formula:

Z = (sample proportion - population proportion) / O(p)

For p = 0.49 + 0.03 (upper limit):

Z = (0.52 - 0.49) / 0.0294 = 1.02

For p = 0.49 - 0.03 (lower limit):

Z = (0.46 - 0.49) / 0.0294 = -1.02

Using the z-table, we find the probabilities corresponding to these z-scores and calculate the difference to get the probability.

The same process is repeated for the scenario where the sample proportion is within ±0.05 of the population proportion to find the respective probability.