Bistro 65 is a chain of Italian restaurants with locations...

Bistro 65 is a chain of Italian restaurants with locations in Ohio and Kentucky. The Bistro 65 menu has four categories of entrées: Pasta, Steak $\&$ Chops, Seafood, and Other (e.g., pizza, sandwiches). Historical data for the chain show that the probability a customer will order an entrée from one of the four categories is .4 for Pasta, .1 for Steak $\&$ Chops, .2 for Seafood, and .3 for Other. A new Bistro 65 restaurant has just opened in Dayton, Ohio, and the following purchase frequencies have been observed for the first 200 customers. a. Conduct a hypothesis test to determine whether the order pattern for the new restaurant in Dayton is the same as the historical pattern for the established Bistro 65 restaurants. Use $\alpha=.05$. b. If the difference in part (a) is significant, prepare a bar chart to show where the differences occur. Comment on any differences observed.

Bistro 65 is a chain of Italian restaurants with locations in Ohio and Kentucky. The Bistro 65 menu has four categories of entrées: Pasta, Steak $\&$ Chops, Seafood, and Other (e.g., pizza, sandwiches). Historical data for the chain show that the probability a customer will order an entrée from one of the four categories is .4 for Pasta, .1 for Steak $\&$ Chops, .2 for Seafood, and .3 for Other. A new Bistro 65 restaurant has just opened in Dayton, Ohio, and the following purchase frequencies have been observed for the first 200 customers.
a. Conduct a hypothesis test to determine whether the order pattern for the new restaurant in Dayton is the same as the historical pattern for the established Bistro 65 restaurants. Use $\alpha=.05$.
b. If the difference in part (a) is significant, prepare a bar chart to show where the differences occur. Comment on any differences observed.

Key Concepts

Chi-Square Goodness-of-Fit Test

The Chi-square goodness-of-fit test is used to determine whether the observed frequencies of categorical data differ significantly from expected frequencies derived from a specified distribution. In this example, it assesses if the observed customer orders align with the historical proportions of the four entrée categories.

Graphical Data Representation (Bar Chart)

Bar charts provide a visual representation of the frequency distribution of categorical variables. In this scenario, a bar chart can be used to compare the observed and expected frequencies across the entrée categories, making it easier to identify which categories significantly contribute to any discrepancies observed.

Significance Level (?)

The significance level, commonly denoted by ?, is the threshold used to decide whether to reject the null hypothesis. In this problem, an ? value of 0.05 means that there is a 5% risk of concluding that a difference exists when there is none.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to assess whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In this context, the null hypothesis typically states that the new restaurant’s order pattern matches the historical order pattern, while the alternative hypothesis suggests that there is a difference.

Expected vs Observed Frequencies

Expected frequencies are the counts predicted by the null hypothesis, calculated by multiplying the total number of observations by the probability of each category from historical data. Comparing these expected frequencies with the actual observed frequencies allows the test to detect deviations from the assumed distribution.

Degrees of Freedom

Degrees of freedom play a crucial role in determining the critical value for the chi-square test. It is calculated as the number of categories minus one, reflecting the constraints imposed by a fixed total sample size and the estimation of parameters.

Transcript

00:01 We'll be performing a kai squared goodness of fit test.

00:05 And i am going to use my software for this.

00:10 And we have the different levels of pasta, steak and chops.

00:15 We have seafood and then other for the meals.

00:18 And we had our data given to us as the 70, 30, 50, 50.

00:27 And we had a model that they believe.

00:30 And their model for i put in list two was that 40 % get pasta, 10 % the steak and chops, 20 % getting the seafood, and 30 % getting the other.

00:44 And we have these totaled up to a total of 200.

00:50 And so my list three, these are my observed, and these will be my expected.

00:56 And i took list two times 200.

01:01 And so those gave me 80, 20, 40, 60.

01:08 And these, again, add up to 200.

01:10 So we will be assuming, oops, we will be assuming that the proportion of the pasta is 0 .4.

01:21 The proportion of steak and chops is equal to 0 .1.

01:26 The proportion of, and let me get rid of this, the proportion of seafood, seafood is equal to 20%, and the proportion of other is equal to 0 .3.

01:40 And then alternately, not all above are correct.

01:49 And i'm going to use my kai squared goodness of fit between these two lists.

01:54 And we can go through and show out the work, but i don't believe you have to do that.

02:01 It doesn't say that you have to show something out.

02:03 So i'm doing a kai square goodness of fit, and that's between my list one and my list three, the way i entered it in...

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