an industry representative claims that 30 percent of all satellite dish owners subscribe to at least one premium movie channels. in an attempt to justify this claim, the representarive's claim is true, and suppose that a sample of six dish owners is randomly selected. asume that one dish owner's permium movie channel subscribition is independent of other dish owners.
To determine whether the industry representative's claim is justified, we can use the concept of probability. Let's break down the problem step by step.
Given information:
- The industry representative claims that 30 percent of all satellite dish owners subscribe to at least one premium movie channel.
- We have a sample of six dish owners.
To analyze this, we need to find the probability that exactly k (where k can range from 0 to 6) dish owners in the sample subscribe to at least one premium movie channel.
First, let's calculate the probability that a single dish owner subscribes to at least one premium movie channel.
The claim states that 30 percent of all satellite dish owners subscribe to at least one premium movie channel. Therefore, the probability of any randomly selected dish owner subscribing is 30 percent, which can be written as 0.30 or 30/100.
Now, using this information, we can calculate the probabilities for different scenarios:
1. Probability that exactly 0 dish owners in the sample subscribe to at least one premium movie channel:
- To calculate this probability, we need to find the probability that a single dish owner does not subscribe to any premium movie channel and then raise it to the power of the sample size.
- The probability that a single dish owner does not subscribe to a premium movie channel is 1 - 0.30 = 0.70 or 70/100.
- Therefore, the probability that exactly 0 dish owners in the sample subscribe to at least one premium movie channel is (0.70)^6.
2. Probability that exactly 1 dish owner in the sample subscribes to at least one premium movie channel:
- To calculate this probability, we multiply the probability that a single dish owner subscribes to a premium movie channel by the probability that the remaining five dish owners do not subscribe.
- This can be written as (0.30)(0.70)^5.
3. Probability that exactly 2 dish owners in the sample subscribe to at least one premium movie channel:
- Similarly to the previous case, we multiply the probability that two dish owners subscribe to a premium movie channel by the probability that the remaining four dish owners do not subscribe.
- In this case, it becomes (0.30)^2(0.70)^4.
4. We can continue this process for scenarios involving 3, 4, 5, and 6 dish owners subscribing to at least one premium movie channel.
Finally, to justify the industry representative's claim, we can compare the calculated probabilities for each scenario with the claim that 30 percent of all satellite dish owners subscribe to at least one premium movie channel. If the calculated probabilities align closely with the claim, it can be considered justified.