Question

It has been observed that a large percentage of professional hockey players have birthdays in the first part of the year. It has been suggested that this is due to the cut-off dates for participation in the youth leagues - those born in the earlier months are older than their peers and this advantage is amplified over the years via more opportunities to train and be coached. Of the 511 professional hockey players in a season, 159 of them were born in January, February, or March.

Requried:
a. Assume that 25% of birthdays from the general population occur in January, February, or March (these actually contain 24.7% of the days of the year). In random samples of 508 people, what is the mean number of those with a birthday in January, February, or March?
b. What is the standard deviation?
c. Now, 158 of the 508 professional hockey players were born in the first three months of the year. With respect to the mean and standard deviation found in parts (a) and (b) what is the z-score for 158?
d. If the men in the professional hockey league were randomly selected from the general population, would 158players out of 508 be an unusual number of men born in the first 3 months of the year?

e. Which of the following is an acceptable sentence to explain this situation?

1. If the men in the professional hockey league were selected randomly from the general population, this would be an unusual collection of birth dates.
2. There is good reason to believe that a significantly larger than expected proportion of professional hockey league players are born in the first three months of the year.
3.This could be a result of the random variation of birth dates within a sample. However, it would be pretty unlikely to happen by chance.
4. All of these are valid statements.

Answer 1

In a random sample of 508 people, the mean number of people with a birthday in January, February, or March is 127. The standard deviation is 11.38. With respect to these statistics, a z-score of 2.72 for 158 players indicates an unusual number, meaning it is unlikely to happen by chance. The acceptable sentence to explain this situation is: All of these are valid statements.

Step-by-step explanation:

a. To calculate the mean number of people with a birthday in January, February, or March in a random sample of 508 people, we can use the formula:

mean = sample size * probability

So, the mean number of people with a birthday in January, February, or March would be:

mean = 508 * 0.25 = 127

b. To calculate the standard deviation, we can use the formula:

standard deviation = square root of (sample size * probability * (1 - probability))

So, the standard deviation would be:

standard deviation = square root of (508 * 0.25 * (1 - 0.25)) = 11.38

c. To calculate the z-score for 158, we can use the formula:

z-score = (x - mean) / standard deviation

So the z-score for 158 would be:

z-score = (158 - 127) / 11.38 = 2.72

d. The z-score of 2.72 indicates that 158 players born in the first three months of the year is an unusually high number compared to the mean and standard deviation found in parts (a) and (b). It is unlikely to happen by chance if the players were randomly selected from the general population.

e. Statement 4 - All of these are valid statements. It is a combination of statements 2 and 3, and acknowledges the possibility of random variation while also recognizing the significance of a larger than expected proportion of professional hockey players born in the first three months of the year.

Answer 2

a

`\mu  =  127`

b

`\sigma  =  9.76`

c

`z-score  =  3.18`

d

Yes, 158 players out of 508 is an unusual number of men born in the first 3 months of the year because the z score of 158 is greater than 3( Note :the probability of z-score = 3 is 97%)

e

The correct option is option 3

Step-by-step explanation:

From the question we are told that

The population proportion is p = 0.25

The sample size is n = 508

Generally the mean is mathematically represented as

`\mu  =  np`

=>
`\mu  =  508 * 0.25`

=>
`\mu  =  127`

Generally the standard deviation is mathematically represented as

`\sigma  =  √( np (1-p))`

=>
`\sigma  =  √( 508 * 0.25 (1-0.25))`

=>
`\sigma  =  9.76`

Generally the z-score of 158 is mathematically represented as

`z-score  =  (158 - 127)/(9.76)`

=>
`z-score  =  (158 - 127)/(9.76)`

=>
`z-score  =  3.18`

Yes, 158 players out of 508 is an unusual number of men born in the first 3 months of the year because the z score of 158 is greater than 3( Note :the probability of z-score = 3 is 97%)

What this means is that the almost the whole professional hockey league player are born in the first month which is unusual