Question

A new muscle relaxant is available. Researchers from the firm developing the relaxant have done studies that indicate that the time lapse between administration of the drug and beginning effect of the drug is normally distributed, with mean = 38 minutes and standard deviation = 5 minutes.

(a) The drug is administered to one patient selected at random. What is the probability that the time it takes to go into effect is 35 minutes or less? (Round your answer to four decimal places.)
________

(b) The drug is administered to a random sample of 10 patients. What is the probability that the average time before it is effective for all 10 patients is 35 minutes or less? (Round your answer to four decimal places.)
________

(c) Comment on the differences of the results in parts (a) and (b).

The probability in part (b) is -
a. the same as
b. larger than
c. smaller than

- the probability in part (a) because the -

a. standard deviation
b. mean

- is -

a. smaller
b. larger

- for the x bar distribution.

Answer 1

To find the probability that the average time before the drug is effective for all 10 patients is 35 minutes or less, use the Central Limit Theorem.

The probability in part (b) is smaller than the probability in part (a) because the standard deviation is smaller for the x bar distribution.

Therefore, the correct options are: c) smaller than, a) standard deviation and a) smaller.

Step-by-step explanation:

To find the probability that the time it takes for the drug to go into effect is 35 minutes or less, we need to calculate the z-score and use the standard normal distribution table.

The z-score formula is z = (x - mean) / standard deviation. Plugging in the values, we have z = (35 - 38) / 5 = -0.6.

Looking up -0.6 in the standard normal distribution table, we find the corresponding probability to be 0.2743.

Therefore, the probability that the time it takes for the drug to go into effect is 35 minutes or less is 0.2743.

To find the probability that the average time before the drug is effective for all 10 patients is 35 minutes or less, we use the Central Limit Theorem.

Since the sample size (n) is greater than 30, we can assume that the sampling distribution of the sample mean is approximately normally distributed.

The formula to calculate the z-score for the sample mean is z = (x - mean) / (standard deviation / sqrt(n)).

Plugging in the values, we have z = (35 - 38) / (5 / sqrt(10)) = -0.9487. Looking up -0.9487 in the standard normal distribution table, we find the corresponding probability to be 0.1715.

Therefore, the probability that the average time before the drug is effective for all 10 patients is 35 minutes or less is 0.1715.

In part (b), the probability is smaller than the probability in part (a) because the standard deviation is smaller for the x bar distribution.