Hello. I would like to ask a college statistics question.

Q. One hundred draws are made at random with replacement from the box [1,2,2,5].
The draws come out as follows:17 [1]'s, 54 [2]'s, 29 [5]'s. Fill in the blanks, using the options below; show work.

(a) For the _______, the observed value is 0.8 SEs above the expected value.

(b) For the _______, the obseved value is 1.33 SEs above the expected value.

Options (one will be left over):
sum of the draws
number of 1's
number of 2's

Well, isn't statistics just a barrel of laughs? Let's see if I can add some funny business to this question!

(a) For the __number of 1's__, the observed value is 0.8 SEs above the expected value. That's quite a 1-derful surprise! It seems like someone's been drawing the short straw or should I say, number.

(b) For the __number of 2's__, the observed value is 1.33 SEs above the expected value. Looks like the number 2 is really having a blast! Maybe they're just too cool for school, or in this case, the box!

Now, where did that number 5 go? It looks like it got lost among all those options. Poor little fella!

Remember, statistics can be a bit tricky, but humor always makes it more fun. Keep crunching those numbers, my friend!

To solve this problem, we need to calculate the expected value and standard error (SE) for each option and then compare them to the observed values.

(a) For the number of 1's:
- Expected value: The probability of drawing a 1 is 1/4, so the expected number of 1's is (1/4) * 100 = 25.
- Standard Error (SE): For a binomial distribution, the SE can be calculated using the formula sqrt(n * p * (1 - p)), where n is the number of draws and p is the probability of success. In this case, n = 100 and p = 1/4. Therefore, the SE is sqrt(100 * 1/4 * (3/4)) = sqrt(75/4) ≈ 4.33.
- Observed value: We have 17 1's, so the observed value is 17.

To determine if the observed value is above the expected value by 0.8 SEs, we calculate:
17 - 25 = -8 (observed - expected) and -8 / 4.33 ≈ -1.85 SEs.
Since the result is negative, the observed value is not 0.8 SEs above the expected value.

(b) For the number of 2's:
- Expected value: The probability of drawing a 2 is 2/4, so the expected number of 2's is (2/4) * 100 = 50.
- Standard Error (SE): Using the same formula as before, the SE is sqrt(100 * 2/4 * (2/4)) = sqrt(100/4) = 5.
- Observed value: We have 54 2's, so the observed value is 54.

To determine if the observed value is above the expected value by 1.33 SEs, we calculate:
54 - 50 = 4 (observed - expected) and 4 / 5 = 0.8 SEs.
The observed value is 0.8 SEs above the expected value.

Therefore, the answers are:
(a) For the number of 1's, the observed value is not 0.8 SEs above the expected value.
(b) For the number of 2's, the observed value is 0.8 SEs above the expected value.

To answer this question, we first need to calculate the expected value and standard error (SE) for each option.

(a) For the blank in part (a), we need to find the option that corresponds to the observed value that is 0.8 standard errors above the expected value.

To calculate the expected value, we can use the formula:

Expected Value = (Probability of outcome 1) * (Number of draws)

In this case, the probability of getting a 1 is 1/4 (since there is only one 1 in the box), and the number of draws is 100.

Expected Value for number of 1's = (1/4) * 100 = 25

Next, we need to calculate the standard error (SE). The formula for SE is the square root of the expected value multiplied by (1 - the probability of outcome).

SE for number of 1's = sqrt(25 * (1 - 1/4)) = sqrt(25 * 3/4) = sqrt(75/4) = sqrt(75)/2 ≈ 4.33

To find the observed value that is 0.8 SEs above the expected value, we multiply the SE by 0.8 and add it to the expected value:

Observed value for number of 1's = 25 + (0.8 * 4.33) ≈ 25 + 3.46 ≈ 28.46

Therefore, the answer for part (a) is "number of 1's."

(b) For the blank in part (b), we need to find the option that corresponds to the observed value that is 1.33 standard errors above the expected value.

Using similar calculations as above:

Expected Value for number of 2's = (2/4) * 100 = 50

SE for number of 2's = sqrt(50 * (1 - 2/4)) = sqrt(50 * 2/4) = sqrt(50)/2 ≈ 3.54

Observed value for number of 2's = 54 + (1.33 * 3.54) ≈ 54 + 4.71 ≈ 58.71

So, the answer for part (b) is "number of 2's."

Therefore, the final answer is:

(a) For the number of 1's, the observed value is 0.8 SEs above the expected value.
(b) For the number of 2's, the observed value is 1.33 SEs above the expected value.

1 should be ¼ so it should be drawn 25 times

2 should be ½ so it should be drawn 50 times
5 should be ¼ so it should be drawn 25 times
Avg=10/4=2.5
S2=(2.25+.25+.25+6.25)/4=9/4=.25 so S=1.5
ES=100(2.5)=250
SE=√100 (1.5)=15
A. number of 2's
B. Sum of the draws (270-250=20 and 20/15=1.33)