Having a lot of trouble with this. Can someone help?

Question

A fine dining restaurant claims that the group sizes of their customers follows the following distribution:

2 people3 people4 people4+ people
32%	13%	18%	37%

Gustavo, a waiter at the restaurant, would like to test this claim. He takes a sample of 97 customers and records the observed frequencies in the following table:

2 people3 people4 people4+ people
41	12	21	23

(a) In performing this statistical test, state the hypotheses.

 H₀: the distribution of customers for each group is not the same as the observed frequencies vs. H_A: the distribution of customers for each group is the same as the observed frequencies

 H₀: the distribution of customers for each group is the same as claimed by the restaurant vs. H_A: the distribution of customers for each group is not the same as claimed by the restaurant

 H₀: the distribution of customers for each group is the same as the observed frequencies vs. H_A: the distribution of customers for each group is not the same as the observed frequencies

 H₀: the proportions of customers for each group are all the same vs. H_A: the proportions of customers for each group are not all the same

 H₀: the distribution of customers for each group is not the same as claimed by the restaurant vs. H_A: the distribution of customers for each group is the same as claimed by the restaurant

(b) What is the expected frequencies of each group? Fill out the table. (Round your answers to 2 decimal places, if needed.)

2 people3 people4 people4+ people

(c) What is the test statistic value for this hypothesis test? (Round your answers to 2 decimal places, if needed.)

TS = 

(d) The test statistic follows a  chi-square distribution with df = 3 chi-square distribution with df = 4 t-distribution with df = 4 chi-square distribution with df = 96 t-distribution with df = 3.

(e) Using the statistical table, the p-value is  0.025 < p-value < 0.05  0.01 < p-value < 0.025  0.05 < p-value < 0.10  0.005 < p-value < 0.01  p-value > 0.10  0 < p-value < 0.005 .

(f) Based on the p-value, those conducting the test should  fail to reject reject the null hypothesis at the significance level of 0.025.

(g) What is the appropriate conclusion?

 There is sufficient evidence to conclude the distribution of customers for each group is the same as claimed by the restaurant.

 There is sufficient evidence to conclude the distribution of customers for each group is not the same as claimed by the restaurant.

 There is insufficient evidence to conclude the distribution of customers for each group is not the same as claimed by the restaurant.

 There is sufficient evidence to conclude the proportions of customers for each group are not all the same.

 There is insufficient evidence to conclude the proportions of customers for each group are not all the same

Answer 1

a)This is a Chi-Squared Goodness of Fit(GOF) Test. H₀: fits the given distribution H_a: does not fit the given distribution

Answer: H₀: the distribution of customers for each group is the same as claimed by the restaurant vs. H_A: the distribution of customers for each group is not the same as claimed by the restaurant

b) The expected counts for a GOF test are found by multiplying the % and the number in the sample.

32%(97)= 31.04

13%(97)= 12.61

18%(97)= 17.46

37%(97)= 35.89

c) You can do this in your calculator by putting the observed values in L₁ and the expected values in L₂ then run a Chi-Squared GOF test . or use the equation the sum of all of the categories (observed-expected)²/expected.

Chi Squared = 3.196+.03+.718+4.629 = 8.573.

d) The df of a GOF test is the number of categories -1.

Therefore this has a df=4-1=3

e)You can then use the Chi-Squared table and look for 8.573 on the table and see what two values it is in between using the correct line for the df of 3. It is in-between 7.81 and 9.35, so its p-value is in between .025 and .05, or if you used a calculator it gives you a p-value of .0355 which is indeed in between .025 and .05

f) If the p-value is less that the significance level, you reject the H_0. If it is greater than the significance level, you fail to reject the H_0.

Since the p-value of .0355 is greater than .025, we would fail to reject the H₀

g). When you reject the H₀ , there is sufficient evidence for the H_a. When you fail to reject the H₀ , there is insufficient evidence for the H_a.

Therefore since we fail to reject,  There is insufficient evidence to conclude the proportions of customers for each group are not all the same.