Question

Determine whether the variable X has a binomial distribution in each of the following cases. If it does, explain why and determine the values of the parameters n and p. If it doesn't, explain why not.

(a) X = Number of days it snows this coming April in Winnipeg.

(b) Weights of apples sold in a grocery store follow a normal distribution with mean 150 grams and standard deviation 20 grams. You take a random sample of five apples.

X = number of apples heavier than 175 grams.

(c) You randomly select a seven-digit phone number from the Winnipeg phone book (excluding area cod

Answer 1

a) no binomial distribution, n is not fixed

Answer 2

`n is not fixed` is not true. The number of days in April is always going to be 30 days

Answer 3

in part (a) X does NOT have a binomial distribution and here's my reasoning why


* it either snows or it doesn't snow. So there are 2 outcomes. Also, n is indeed fixed to n = 30. These facts are true but the other facts aren't so clear cut.

* each day is independent from the other days (though I'm not a meteorologist so this claim may not be 100% true?)

* the probability that it snows varies over time. It would be nice if every day had a fixed chance of snow so you could plan, but life isn't that simple. So because p is NOT fixed, this means that we do NOT have a binomial distribution

Answer 4

okay thank you. for b) it is a binomial. n=5

Answer 5

what is p in part (b) ?

Answer 6

wait i think it shud be this instead: This is not a binomial. Though n is fixed, n=5, there are more than 2 possible outcomes. Also there is no probability of success.

using x- µ/σ/√n =175-150/20/√5 =2.795

Answer 7

`there are more than 2 possible outcomes` not true. You either have an apple lighter or of equal weight to 175 grams OR you have an apple that is heaver than 175 grams


outcome 1: lighter or of equal weight to 175 grams
OR
outcome 2: heaver than 175 grams

Answer 8

`there is no probability of success.` yes there is. Determine the area under the curve to the right of x = 175 to find the value of p

Answer 9

under the normal distribution curve

Answer 10

I should note that I'm operating under the assumption that there are so many apples that it's almost as if there is replacement going on

Answer 11

okay! c) This is not a binomial. There is no fixed number of trials. Though there are only two outcomes; getting a zero or a non zero digit, and the probability of success is constant

Answer 12

part (c) looks cut off based on what you posted at the very top

Answer 13

okay sorry.. let me post the full questions:
(c) You randomly select a seven-digit phone number from the Winnipeg phone book (excluding area code).

X = number of zeroes in the phone number.

(d) The number of goals scored by the Winnipeg Jets follows a Poisson distribution with a rate of 2.6 per game. We take a random sample of ten Jets games.

X = number of games in which the Jets are shutout (i.e., they score zero goals).

The remainder of the questions refer to the following:

Suppose we have three boxes, each containing a number of coloured balls:

Box 1 contains 5 red balls, 3 yellow balls and 2 green balls.
Box 2 contains 9 red balls, 6 yellow balls and 5 green balls.
Box 3 contains 14 red balls, 9 yellow balls and 7 green balls.

Answer 14

(e) You randomly select 5 balls from Box 1 with replacement.

X = number of red balls selected.

(f) You randomly select 5 balls from Box 2 without replacement.

X = number of green balls selected.

(g) You randomly select balls from Box 3, with replacement, until the second yellow ball is selected.

X = number of trials to get the second yellow ball.

(h) You randomly select one ball from each box.

X = number of red balls selected.

(i) You randomly select one ball from each box.

X = number of yellow balls selected.

Answer 15

`(c) You randomly select a seven-digit phone number from the Winnipeg phone book (excluding area code).`

` X = number of zeroes in the phone number.`

are you sure that n is not fixed?

Answer 16

i think it is not fixed, because we really dont know how many zeroes would a phone number contain

Answer 17

but the sample size is fixed. You can easily determine the number of phone numbers in the book. Sure it would take a while, but it's information that can be figured out

Answer 18

so for c)This is a binomial. There is fixed number of trials. Though there are only two outcomes; getting a zero or a non zero digit, and the probability of success is constant

Answer 19

what would p be in part (c) ?

Answer 20

that i do not know how to obtain

Answer 21

hmm I'm not sure either and I'm starting to think (c) isn't binomial because of how it's stated

Answer 22

so is it because of the probability of success?

Answer 23

well how do you define a "success"? a number that has at least one zero? that's not really clear

Answer 24

can we jump to d first, while i figure out c

Answer 25

c. This is not a binomial. There is no fixed number of trials. Though there are only two outcomes; getting a zero or a non zero digit, and the probability of success is constant

d. This is a Binomial. There is a fixed number if trials, n=10. There are only two possible outcomes. Either the jets score have zero goals or they score more than zero goals. Each of the 10 games are independent of each other. And lastly the probability of success is constant form game to game; given as λ =2.62 =0.29

P(x=10); where λ =2.6

2.610 e-2.6/10! =14116.7096*0.07427/3628800 =0.29

e. This is not a Binomial. Since, we are taking a random sample of 5, it goes to say that the probability of success (the number of red balls selected)can not be determined from sample to sample, even though the balls are replaced as they are randomly selected.

f. This is not a binomial. The balls are selected without replacement, which means that the probability of success is not constant across trials.

g. This is not a binomial. Even though you select balls form box 3 with replacement, ensuring that the probability of success is constant, The number of times one should select the ball is not specified. That is there is no fixed number of trials.

h. If you randomly select one ball form each box, without replacement, the probability of success across trials is not constant. Also there is no fixed number of trials for the item of interest (number of red balls)

I. This is not a binomial, as the number of times one should select ball from each box is not specified. n is not fixed in advance of the experiment, though the outcome of interest is specified.
a

Answer 26

(d) says poisson and not binomial. I'm reading that a poisson is approximately a binomial if n gets really really large while p gets really really small

Answer 27

so I'm thinking D isn't binomial because of that

Answer 28

hmm then again n is fixed at n = 10 and the probability could be fixed. What that exact probability is, I don't know. I'm not so sure about these problems. Sorry