An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Define populations: All ratings for Chalio's vs. all ratings for ordinary restaurants.
State hypotheses: H 0 : Uniform distribution; H 1 : Non-uniform distribution.
Calculate the Chi-Square statistic: χ 2 = 4 .
Compare to the critical value: Since 4 < 6.745 , fail to reject H 0 . The ratings are uniformly distributed. Fail to reject H 0
Explanation
Define Populations First, let's define the populations for our hypothesis test.
Population 1: The set of all star ratings reported on Chalio's Restaurant made by local customers. Population 2: The set of all star ratings reported on ordinary or static restaurants made by local customers.
State Hypotheses Now, let's state the null and alternative hypotheses.
Null Hypothesis ( H 0 ): The star ratings for Chalio's are uniformly distributed. This means that each star rating (1 to 5) has an equal chance of being selected.
Alternative Hypothesis ( H 1 ): The star ratings for Chalio's are not uniformly distributed. This suggests that Chalio's is either loved or disliked, leading to a non-uniform distribution of star ratings.
Verify Requirements Next, we need to check the requirements for the Chi-Square Goodness-of-Fit test. The primary requirement is that each expected frequency is at least 5. In this case, the expected frequency for each category is 20, which is greater than 5, so the requirement is met.
Calculate Chi-Square Statistic Now, let's calculate the Chi-Square test statistic. We have the following observed frequencies:
5 Stars: 21 4 Stars: 15 3 Stars: 18 2 Stars: 19 1 Star: 27
The expected frequency for each category under the null hypothesis (uniform distribution) is 5 100 = 20 .
The Chi-Square test statistic is calculated as:
χ 2 = ∑ E i ( O i − E i ) 2
where O i is the observed frequency and E i is the expected frequency for each category.
χ 2 = 20 ( 21 − 20 ) 2 + 20 ( 15 − 20 ) 2 + 20 ( 18 − 20 ) 2 + 20 ( 19 − 20 ) 2 + 20 ( 27 − 20 ) 2
χ 2 = 20 1 + 20 25 + 20 4 + 20 1 + 20 49 = 20 80 = 4
Calculate Degrees of Freedom The degrees of freedom (df) for the Chi-Square test are calculated as:
df = number of categories - 1 = 5 - 1 = 4
Determine Critical Value and P-value We are given a significance level of α = 0.15 . We need to find the critical value for the Chi-Square distribution with 4 degrees of freedom and a significance level of 0.15. From the Chi-Square distribution table or using a calculator, the critical value is approximately 6.745.
We also calculate the p-value associated with the test statistic of 4 and 4 degrees of freedom. The p-value is approximately 0.406.
Compare Test Statistic and Critical Value Now, we compare the calculated Chi-Square test statistic (4) to the critical value (6.745). Since 4 < 6.745, we fail to reject the null hypothesis.
Alternatively, we can compare the p-value (0.406) to the significance level (0.15). Since 0.406 > 0.15, we also fail to reject the null hypothesis.
Interpret Results and Conclusion Since we fail to reject the null hypothesis, we conclude that there is not enough evidence to suggest that the star ratings for Chalio's are not uniformly distributed. Therefore, based on this analysis, we cannot conclude that Chalio's is either loved or disliked at the 15% significance level.
Examples
Imagine you're analyzing customer satisfaction for a new product. You collect ratings on a scale of 1 to 5 and want to know if the ratings are uniformly distributed (indicating average satisfaction) or skewed (indicating strong positive or negative sentiment). This Chi-Square test helps you determine if the observed distribution of ratings significantly differs from a uniform distribution, guiding your marketing and product improvement strategies.
