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Skittles Statistics teacher Jason Molesky contacted Mars, Inc., to ask about the color distribution for Skittles candies. Here is an excerpt from the response he received: "The original flavor blend for the SKITTLES BITE SIZE CANDIES is lemon, lime, orange, strawberry and grape. They were chosen as a result of consumer preference tests we conducted. The flavor blend is 20 percent of each flavor." (a) State appropriate hypotheses for a significance test of the company's claim. (b) Find the expected counts for a bag of Skittles with 60 candies. (c) How large a \(\chi^{2}\) statistic would you need to have significant evidence against the company's claim at the \(\alpha=0.05\) level? At the \(\alpha=0.01\) level? (d) Create a set of observed counts for a bag with 60 candies that gives a \(P\) -value between 0.01 and \(0.05 .\) Show the calculation of your chi-square statistic.

Step by step solution

01

Formulate the Hypotheses

To test the company's claim about the distribution of Skittles colors, we use the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). - \(H_0\): Each color of Skittle (lemon, lime, orange, strawberry, grape) has a population proportion of 0.20.- \(H_a\): Not all colors have a population proportion of 0.20.

02

Calculate Expected Counts

The expected count for each color in a bag of 60 Skittles, given the company's claim, is calculated as follows:- Total candies = 60- Proportion for each color = 0.20Expected count for each color: \[ 0.20 \times 60 = 12 \]

03

Determine Critical Values for Chi-Square

We use the chi-square distribution table to find the critical values for significance levels \(\alpha = 0.05\) and \(\alpha = 0.01\). - \(df = k - 1 = 5 - 1 = 4\) where \(k\) is the number of categories.- Critical value for \(\alpha = 0.05\): \(\chi^2_{0.05, 4} = 9.488\) - Critical value for \(\alpha = 0.01\): \(\chi^2_{0.01, 4} = 13.277\)

04

Create Observed Counts and Calculate Chi-Square Statistic

Create a set of observed counts such that the \(P\)-value is between 0.01 and 0.05. Here is an example:Observed counts: {[10, 13, 14, 11, 12]}Calculate the \(\chi^2\) statistic:\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]Where \(O_i\) are the observed counts and \(E_i\) are the expected counts (12 for each color).\[\chi^2 = \frac{(10-12)^2}{12} + \frac{(13-12)^2}{12} + \frac{(14-12)^2}{12} + \frac{(11-12)^2}{12} + \frac{(12-12)^2}{12} = \frac{4}{12} + \frac{1}{12} + \frac{4}{12} + \frac{1}{12} + \frac{0}{12} = 1.00\]This chi-square statistic should ideally be between the critical values found in Step 3 (i.e., create different observed counts if necessary) to get a \(P\)-value between 0.01 and 0.05.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Hypotheses

When conducting a Chi-Square Test on Skittles flavors, statistical hypotheses play a crucial role.
Hypotheses are statements that we propose to evaluate using statistical data.

In this context, we use two types of hypotheses: the null hypothesis and the alternative hypothesis.

• **Null Hypothesis ( \(H_0\) )**: This hypothesis states that each color of Skittles (lemon, lime, orange, strawberry, grape) makes up 20% of the total population of Skittles. Thus, mathematically expressed as: each color's proportion = 0.20.
• **Alternative Hypothesis ( \(H_a\) )**: Contrary to the null hypothesis, it suggests that one or more Skittle colors do not have the claimed 20% distribution.

These hypotheses are critical as they set the framework for the Chi-Square Test and determine the direction of our test analysis. Success or failure in rejecting the null hypothesis will indicate whether Mars, Inc.'s claim about Skittles' color distribution is supported by the test data or not.
By clearly defining these hypotheses, statisticians can objectively evaluate if the observed sample data aligns with the expected distribution in the hypothesis.

Expected Counts

In a Chi-Square Test, expected counts are vital calculations that help us determine if observed data significantly deviates from what we would expect. To compute expected counts for the Skittles, we base our calculation on Mars, Inc.'s claim that each color is equally distributed at 20% in every packet.

• **Total candies**: For a bag containing 60 Skittles, each color should theoretically represent 20% or one-fifth of the total count.
• **Expected count per color**: Mathematically, we multiply the proportion by the total candies. Hence, the expected count becomes: \[0.20 \times 60 = 12\]

This expected count of 12 Skittles per color serves as a baseline for comparison with actual sample observations.
Deviations from this expected count help us determine the variance in the Chi-Square Test. It's important because this comparison tells us if any discrepancies are due to random chance or suggest a different distribution pattern among the Skittles colors.
By understanding and calculating the expected counts, we can delve deeper into assessing whether the claimed distribution aligns with our sample data.

Significance Level

The significance level, denoted by \(\alpha\), is a threshold used in hypothesis testing to determine statistical significance.
It helps us decide whether to accept the null hypothesis or not.

• Common significance levels include **0.05** and **0.01**, representing a 5% and 1% risk of concluding that a difference exists when there is none.
• These levels act as benchmarks; a result falling below \(\alpha\) suggests significant evidence against the null hypothesis.

In this Skittles exercise, finding the critical values at these significance levels helps identify the size of the Chi-Square statistic needed to find significant evidence against Mars, Inc.'s claim about color distribution.Using a chi-square distribution table:

• For \(\alpha = 0.05\) with \(df = 4\), the critical value is approximately 9.488.
• For \(\alpha = 0.01\), it is around 13.277.

If the calculated Chi-Square statistic exceeds these values, we reject the null hypothesis.
Understanding these levels allows researchers, like ourselves, to assess whether Mars, Inc.'s color distribution claims for Skittles are statistically plausible.
It's a key aspect of data analysis, with the final decision hinging on whether our test statistic falls within or outside these critical threshold values.