An article studied differences between expert and consumer ratings by considering medal ratings for wines, which could be gold (G), silver (S), or bronze (B). Three categories were then established. 1. Rating is the same [(G, G), (B, B), (S, S)] 2. Rating differs by one medal [(G, S), (S, G), (S, B), (B, S)] 3. Rating differs by two medals [(G, B), (B, G)] The observed frequencies for these three categories were 66, 104, and 42, respectively. On the hypothesis of equally likely expert ratings and consumer ratings being assigned completely by chance, each of the nine medal pairs has probability 1/9. Carry out an appropriate chi-squared test using a significance level of 0.10.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![An article studied differences between expert and consumer ratings by considering medal ratings for wines, which could be gold (G), silver (S), or bronze (B). Three categories were then established:
1. Rating is the same [(G, G), (B, B), (S, S)]
2. Rating differs by one medal [(G, S), (S, G), (S, B), (B, S)]
3. Rating differs by two medals [(G, B), (B, G)]
The observed frequencies for these three categories were 66, 104, and 42, respectively. On the hypothesis of equally likely expert ratings and consumer ratings being assigned completely by chance, each of the nine medal pairs has a probability of 1/9. Carry out an appropriate chi-squared test using a significance level of 0.10.
State the appropriate hypotheses.
- \(H_0: p_1 = p_2 = p_3 = \frac{1}{3}\)
- \(H_a: \text{ at least one } p_i \neq \frac{1}{3}\)
- \(H_0: p_1 = \frac{1}{3}, p_2 = \frac{2}{9}, p_3 = \frac{4}{9}\)
- \(H_a: \text{ at least one } p_i \neq p_{i0}\)
- \(H_0: p_1 = p_2 = p_3 = \frac{1}{9}\)
- \(H_a: \text{ at least one } p_i \neq \frac{1}{9}\)
- \(H_0: p_1 = \frac{1}{3}, p_2 = \frac{4}{9}, p_3 = \frac{2}{9}\)
- \(H_a: \text{ at least one } p_i \neq p_{i0}\)
Calculate the test statistic. (Round your answer to two decimal places.)
\[ \chi^2 = \_\_\_\_ \]
What can be said about the \(P\)-value for the test?
- \(P\)-value < 0.005
- 0.005 < \(P\)-value < 0.01
- 0.01 < \(P\)-value < 0.025
- 0.025 < \(P\)-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda5cf1ef-ed5b-45b4-b720-27437f28fd1b%2Fc2702f0f-007f-4bab-9385-5f99391bfea0%2F4yx1bsi_processed.jpeg&w=3840&q=75)
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