Math  /  Data & Statistics

QuestionQuestion 3 【14]
Consider the following cross- tabulation of brand preference for digital cameras and their primary usage (professional or personal): \begin{tabular}{|c|c|c|c|c|} \hline \multirow{2}{*}{ Usage } & \multicolumn{3}{|c|}{ Camera Brand } & Total \\ \cline { 2 - 5 } & Nixon & Canon & Pentax & \\ \hline Professional & 46 & 42 & 64 & 152 \\ \hline Personal & 54 & 60 & 46 & 150 \\ \hline Total & 100 & 102 & 110 & 312 \\ \hline \end{tabular} (a) What is the probability of selecting a professional user? (3) (b) Find the probability that a user prefers the Pentax brand given that their usage is primarily personal. (4) (c) Find the probability of randomly selecting either a professional user or a user who prefers the Nikon brand of digital camers or both. (3) (d) Are the two events primary usage and brand preference mutually exclusive? (2) Justify your answer statistically (e) What is the likelihood that a randomly selected user prefers the Canon trand and is a professional user?

Studdy Solution

STEP 1

1. The total number of respondents is 312.
2. The cross-tabulation provides counts for different combinations of camera brand preference and primary usage.
3. Probabilities are calculated as the ratio of favorable outcomes to the total number of respondents.

STEP 2

1. Calculate the probability of selecting a professional user.
2. Calculate the conditional probability of a user preferring the Pentax brand given that their usage is primarily personal.
3. Calculate the probability of randomly selecting either a professional user, a user who prefers the Nikon brand, or both.
4. Determine if the events "primary usage" and "brand preference" are mutually exclusive.
5. Calculate the likelihood that a randomly selected user prefers the Canon brand and is a professional user.

STEP 3

Calculate the probability of selecting a professional user.
The number of professional users is 152, and the total number of respondents is 312. P(Professional)=152312 P(\text{Professional}) = \frac{152}{312}

STEP 4

Simplify the fraction to get the probability. P(Professional)=152312=38780.487 P(\text{Professional}) = \frac{152}{312} = \frac{38}{78} \approx 0.487

STEP 5

Calculate the probability that a user prefers the Pentax brand given that their usage is primarily personal.
The number of personal users who prefer Pentax is 46, and the total number of personal users is 150. P(Pentax | Personal)=46150 P(\text{Pentax | Personal}) = \frac{46}{150}

STEP 6

Simplify the fraction to get the conditional probability. P(Pentax | Personal)=461500.307 P(\text{Pentax | Personal}) = \frac{46}{150} \approx 0.307

STEP 7

Calculate the probability of selecting either a professional user or a user who prefers the Nikon brand or both.
First, find the probability of selecting a user who prefers the Nikon brand. The number of users who prefer Nikon is 100, and the total number of respondents is 312. P(Nikon)=100312 P(\text{Nikon}) = \frac{100}{312}

STEP 8

Simplify the fraction to get the probability. P(Nikon)=1003120.321 P(\text{Nikon}) = \frac{100}{312} \approx 0.321

STEP 9

Use the principle of inclusion and exclusion to find the probability of selecting either a professional user, a user who prefers the Nikon brand, or both.
We need to subtract the probability of the intersection of the two events to avoid double-counting. The number of professional users who prefer the Nikon brand is 46. P(ProfessionalNikon)=P(Professional)+P(Nikon)P(ProfessionalNikon) P(\text{Professional} \cup \text{Nikon}) = P(\text{Professional}) + P(\text{Nikon}) - P(\text{Professional} \cap \text{Nikon}) P(ProfessionalNikon)=46312 P(\text{Professional} \cap \text{Nikon}) = \frac{46}{312}

STEP 10

Simplify the fraction to get the intersection probability. P(ProfessionalNikon)=463120.147 P(\text{Professional} \cap \text{Nikon}) = \frac{46}{312} \approx 0.147

STEP 11

Combine the probabilities to get the final probability using inclusion-exclusion principle. P(ProfessionalNikon)=0.487+0.3210.147=0.661 P(\text{Professional} \cup \text{Nikon}) = 0.487 + 0.321 - 0.147 = 0.661

STEP 12

Determine if the events "primary usage" and "brand preference" are mutually exclusive.
Two events are mutually exclusive if they cannot occur at the same time. Since there are professional users who prefer different brands, and personal users who prefer different brands, these events are not mutually exclusive.

STEP 13

Calculate the likelihood that a randomly selected user prefers the Canon brand and is a professional user.
The number of professional users who prefer Canon is 42, and the total number of respondents is 312. P(CanonProfessional)=42312 P(\text{Canon} \cap \text{Professional}) = \frac{42}{312}

STEP 14

Simplify the fraction to get the probability. P(CanonProfessional)=423120.135 P(\text{Canon} \cap \text{Professional}) = \frac{42}{312} \approx 0.135
Solution: (a) The probability of selecting a professional user is approximately 0.487. (b) The probability that a user prefers the Pentax brand given that their usage is primarily personal is approximately 0.307. (c) The probability of randomly selecting either a professional user or a user who prefers the Nikon brand or both is approximately 0.661. (d) The events "primary usage" and "brand preference" are not mutually exclusive because there are users with each usage type who prefer different brands. (e) The likelihood that a randomly selected user prefers the Canon brand and is a professional user is approximately 0.135.

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