A study has been made of the relationship between weight gain and fast food cons

Question

A study has been made of the relationship between weight gain and fast food consumption. The participants were classified in four categories of weight change (lose weight / do not change / gain little weight / gain a lot of weight.) The odds ratio of not being a conditional fast food consumer to having lost weight is 9. The probability of gaining little weight and gaining a lot of weight both conditional on being a fast food consumer are 0.3 and 0.2 respectively.The odds ratio of not being a fast food consumer conditional on not having changed weight is 3 and conditional on having gained a lot of weight is 0.2. The marginal probability of losing weight is 0.1. The marginal probability of being a fast food consumer is 0.3
(a) Obtain the frequency table (b) Calculate the marginal probability of gaining a lot of weight and the marginal probability of not being a consumer of fast food (c) The probability of being a conditional food fast consumer to gain a lot of weight (d) The marginal probability of gaining little or a lot of weight (e) The odds ratio of gaining a lot of weight conditional to be a fast food consumer (f) The probability of losing weight conditional of not being a consumer of fast food (g) The probability of gaining a lot of weight and not being a consumer of fast food A study has been made of the relationship between weight gain and fast food consumption. The participants were classified in four categories of weight change (lose weight / do not change / gain little weight / gain a lot of weight.) The odds ratio of not being a conditional fast food consumer to having lost weight is 9. The probability of gaining little weight and gaining a lot of weight both conditional on being a fast food consumer are 0.3 and 0.2 respectively.The odds ratio of not being a fast food consumer conditional on not having changed weight is 3 and conditional on having gained a lot of weight is 0.2. The marginal probability of losing weight is 0.1. The marginal probability of being a fast food consumer is 0.3
(a) Obtain the frequency table (b) Calculate the marginal probability of gaining a lot of weight and the marginal probability of not being a consumer of fast food (c) The probability of being a conditional food fast consumer to gain a lot of weight (d) The marginal probability of gaining little or a lot of weight (e) The odds ratio of gaining a lot of weight conditional to be a fast food consumer (f) The probability of losing weight conditional of not being a consumer of fast food (g) The probability of gaining a lot of weight and not being a consumer of fast food A study has been made of the relationship between weight gain and fast food consumption. The participants were classified in four categories of weight change (lose weight / do not change / gain little weight / gain a lot of weight.) The odds ratio of not being a conditional fast food consumer to having lost weight is 9. The probability of gaining little weight and gaining a lot of weight both conditional on being a fast food consumer are 0.3 and 0.2 respectively.The odds ratio of not being a fast food consumer conditional on not having changed weight is 3 and conditional on having gained a lot of weight is 0.2. The marginal probability of losing weight is 0.1. The marginal probability of being a fast food consumer is 0.3
(a) Obtain the frequency table (b) Calculate the marginal probability of gaining a lot of weight and the marginal probability of not being a consumer of fast food (c) The probability of being a conditional food fast consumer to gain a lot of weight (d) The marginal probability of gaining little or a lot of weight (e) The odds ratio of gaining a lot of weight conditional to be a fast food consumer (f) The probability of losing weight conditional of not being a consumer of fast food (g) The probability of gaining a lot of weight and not being a consumer of fast food

Explanation / Answer

For simplicity, Let us first assign a variable to each event.
Person being a fast food consumer: FC
Person of not being a fast food consumer: NFC
Person who lose weight: A
Person whose weight did not change: B
Person who gained little weight: C
Person who gained a lot of weight: D

Given that:
Odds ratio (NFC / A) = 9
P(C | FC)=0.3
P(D | FC)=0.2
Odds ratio (NFC/B)= 3
Odds ratio (NFC/D)= 0.2
P(A)=0.1
P(FC)=0.3

To find:
(b) P(D) & P(NFC)
(c) P(FC & D)
(d) P(C or D)
(e) Odds ratio (D / FC)
(f) P(A | FC)
(g) P(C and NFC)

Now, Odds ratio (NFC / A) = Odd (NFC) / Odd (A) = 9
Odd (A) = P(A) / (1- P(A)
= 0.1 / (1-0.1)
= 1/9
Putting in above equation:
Odd(NFC) = 1 >>>> Ref 1
i.e P(NFC) / (1- P(NFC) = 1
P(NFC) = 0.5

P(C | FC) = P (C and FC) / P(FC) = 0.3
P (C and FC ) = 0.3 * 0.3 = 0.09 [ P(FC) = 0.3]
P( D| FC) = P (D and FC) / P(FC) = 0.2
P (D and FC) = 0.2 * 0.3 = 0.06

Odds ratio ( NFC / B) = Odd (NFC) / Odd (B) = 3
Odd (B) = 1 / 3 [ Ref 1 : Odd(NFC) = 1]
P(B) / (1- P(B)) = 1/3
P(B) = 0.25

Odds ratio ( NFC / D) = Odd (NFC) / Odd (D) = 0.2
Odd (D) = 1/0.2[ Ref 1 : Odd(NFC) = 1]
Odd(D) = 5
P(D)/ (1-P(D)) = 5
P(D) = 1/6 = 0.16
Now, P(A) + P(B) + P(C) + P(D) = 1
0.1 + 0.25 + P(C) + 0.16 = 1
P(C) = 0.49

(b) P(D) = 1/6 & P(NFC) = 0.5
(c) P(D and FC) = 0.06
(d) P( C or D) = P(C) * P(D)
= 0.49 * 0.16
= 0.0784
(e) Odds ratio ( D/ FC) = Odd (D) / Odd(FC)
Odd (D) = 5 (as calculated above)
Odd(FC) = P(FC) / (1- P(FC)
= 0.3 / (1-0.3)
= 3/7
Odds ratio ( D/ FC) = 5/ (3/7)
= 35/3
(g) P ( C and NFC) = P(C) * P(NFC)
= 0.49* 0.5
= 0.245

This is the approach to solve this problem, however, there seems to be an error in this question with regards to the parameter and logic.

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