Plain M&M's come in 6 different colors (Blue, Orange, Green, Yellow, Red, Brown) and are produced at two different plants. M&M's that come from a plant in Tennessee are supposed to have the following distribution of colors: 20.7% Blue; 20.5% Orange; 19.8% Green; 13.5% Yellow; 13.1% Red and 12.4% Brown. Quality control at the plant is concerned the machine is not working correctly and that it is producing a different distribution of colors. They take a random sample of 940 plain M&M's. A Chi-Square Goodness of Fit Test is performed. Calculate the value of the chi-square goodness of fit test statistic given the observed counts below. Enter your answer to 3 decimal places. You will need to enter the expected count for Blue from the previous problem; use the value to calculate the contribution to chi-square for the Blue category and total the last column to get the overall test statistic value. Color Observed Count Expected Count Contribution to the Chi-Square test statistic = (Observed - Expected)2/Expected Blue 185 Orange 195 192.70 (195-192.7)2/192.70 = 0.027 Green 180 186.12 (180-186.12)2/186.12 = 0.201 Yellow 140 126.90 (140-126.90)2/126.90 = 1.352 Red 121 123.14 (121-123.14)2/123.14 = 0.037 Brown 119 116.56 (119-116.56)2/116.56 = 0.051 Total 940
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.
Plain M&M's come in 6 different colors (Blue, Orange, Green, Yellow, Red, Brown) and are produced at two different plants. M&M's that come from a plant in Tennessee are supposed to have the following distribution of colors: 20.7% Blue; 20.5% Orange; 19.8% Green; 13.5% Yellow; 13.1% Red and 12.4% Brown. Quality control at the plant is concerned the machine is not working correctly and that it is producing a different distribution of colors. They take a random sample of 940 plain M&M's. A Chi-Square Goodness of Fit Test is performed. Calculate the value of the chi-square goodness of fit test statistic given the observed counts below. Enter your answer to 3 decimal places. You will need to enter the expected count for Blue from the previous problem; use the value to calculate the contribution to chi-square for the Blue category and total the last column to get the overall test statistic value.
| Color |
Observed Count |
Expected Count |
Contribution to the Chi-Square test statistic = (Observed - Expected)2/Expected |
| Blue | 185 | ||
| Orange | 195 | 192.70 | (195-192.7)2/192.70 = 0.027 |
| Green | 180 | 186.12 | (180-186.12)2/186.12 = 0.201 |
| Yellow | 140 | 126.90 | (140-126.90)2/126.90 = 1.352 |
| Red | 121 | 123.14 | (121-123.14)2/123.14 = 0.037 |
| Brown | 119 | 116.56 | (119-116.56)2/116.56 = 0.051 |
| Total | 940 |
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