Question

The mean incubation time for a type of fertilized egg kept at a certain temperature is 20 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 2 days.

Part A

Find and interpret the probability that a randomly selected fertilized egg hatched in less than 18 days.

A. 15.87%
B. 13.5%
C. 1.69%
D. 6.68%
Interpretation: If 100 fertilized eggs were randomly selected, approximately 16 of them would be expected to hatch in less than 18 days.

Part B

Find and interpret the probability that a randomly selected fertilized egg takes over 22 days to hatch.

A. 15.87%
B. 13.5%
C. 1.69%
D. 6.68%
Interpretation: If 100 fertilized eggs were randomly selected, approximately 16 of them would be expected to take over 22 days to hatch.

Part C

Find and interpret the probability that a randomly selected fertilized egg hatched between 16 and 20 days.

A. 68.27%
B. 13.5%
C. 1.69%
D. 15.87%
Interpretation: If 100 fertilized eggs were randomly selected, approximately 68 of them would be expected to hatch between 16 and 20 days.

Part D

Would it be unusual for an egg to hatch in less than 14 days?

A. Yes, because the probability of an egg hatching in less than 14 days is 0.135%.
B. No, because the probability of an egg hatching in less than 14 days is 6.68%.
C. Cannot be determined

Answer 1

Using the concept of normal distribution, the probabilities of incubation times for fertilized eggs are found. The correct probabilities are 15.87% for both less than 18 days and more than 22 days to hatch, and 68.27% for hatching between 16 and 20 days. It would be unusual for an egg to hatch in less than 14 days.

Step-by-step explanation:

The probability questions you've asked about are based on the concept of normal distribution, which is used in statistics to describe how data is spread around a mean value. In this case, you're looking at the incubation times of fertilized eggs.

Part A

To find the probability that an egg hatched in less than 18 days, we calculate the z-score for 18 days and then refer to a standard normal distribution table or use a calculator with statistical functions. Here, the z-score is (18 - 20) / 2 = -1. Using the cumulative normal distribution function or table, we find that approximately 15.87% of eggs would hatch in less than 18 days, making option A correct for both probability and interpretation.

Part B

Similarly, for eggs that take over 22 days to hatch, the z-score is (22 - 20) / 2 = 1. Referring again to the normal distribution, we would find about 15.87% of eggs would take more than 22 days to hatch, making option A correct again for both probability and interpretation.

Part C

To find out how many eggs hatch between 16 and 20 days, we need the combined probabilities of z-scores from -2 to 0. This area under the standard normal curve is approximately 68.27%, making option A the correct choice and the interpretation that about 68 out of 100 eggs would hatch in this time frame accurate.

Part D

For an egg to hatch in less than 14 days, we need the z-score for 14 days which is (14 - 20) / 2 = -3. This is quite far on the left tail of the normal distribution, making it a very low probability event, and indeed it would be unusual; however, without the exact probability, we cannot choose between the given options. Thus, it can be said that it's highly unlikely, but a definitive option cannot be selected from the given answers without further computation.