A market research group is interested in estimating the market share of three airline companies, A, B, and C, operating on a particular flight path. These three airlines are the only companies operating on this route. Airline B claims that they have 50% of the business, with the remaining percentage shared equally amongst airlines A and C, i.e., 25% each.

The market research group decides to carry out an independent study and question a random sample of 69 people visiting an airport on a particular day. Each person was given price, timetables, and service details of flights on this route for each of the three airlines. Provided with this information, they were asked to select their airline of choice. The survey revealed the following observed frequencies:
- Airline C: 25
- Airline A: 17
- Airline B: 27

Test if there is any evidence to suggest that the airline's claim is incorrect, at significance level [tex]\alpha = 0.05[/tex].

Complete the test by answering the following questions:

1. Which of the following is the correct choice of hypotheses for the test:
a) [tex]H_0: p_1 = p_2 = p_3 = 0.33[/tex] versus [tex]H_a: \text{at least one } p_i \neq 0.33[/tex]
b) [tex]H_0: p_1 = 0.25, p_2 = 0.5, p_3 = 0.25[/tex] versus [tex]H_a: p_1 \neq 0.25, p_2 \neq 0.5, p_3 \neq 0.25[/tex]
c) [tex]H_0: p_1 = 0.25, p_2 = 0.5, p_3 = 0.25[/tex] versus [tex]H_a: \text{at least one } p_i \text{ not as specified in the null hypothesis}[/tex]
d) [tex]H_0: p_1 = p_2 = p_3 = 0.33[/tex] versus [tex]H_a: p_1 \neq p_2 \neq p_3 \neq 0.33[/tex]
Insert your answer (a, b, c, or d).

2. Assuming the null hypothesis is true, the frequencies you would expect to see in the sample are:
a) [tex]E_1 = 17[/tex], [tex]E_2 = 27[/tex], [tex]E_3 = 25[/tex]
b) [tex]E_1 = 23[/tex], [tex]E_2 = 23[/tex], [tex]E_3 = 23[/tex]
c) [tex]E_1 = 17.25[/tex], [tex]E_2 = 34.5[/tex], [tex]E_3 = 17.25[/tex]
Insert your answer (a, b, or c).

3. Fill in the following blanks:
- The test statistic has value TS = __________.
- The distribution is (examples: normal / [tex]t_{12}[/tex] / [tex]\chi^2_{4}[/tex] / [tex]F_{5,6}[/tex]).
- Testing at significance level [tex]\alpha = 0.05[/tex], the rejection region is: (less/greater) than (2 dec places).
- Since the test statistic (is in/is not in) the rejection region, there (is evidence/is no evidence) to reject the null hypothesis, [tex]H_0[/tex].

4. Which of the following is the correct conclusion for this test:
a) There is no evidence to reject Airline B's claimed values of market share proportions.
b) There is evidence to reject Airline B's claimed values of market share proportions.
Insert your answer (a or b).

5. When checking the validity of any assumptions required for the inference made here to be valid, which of the following is the correct statement:
a) The sample size is large enough since the total size is greater than 30.
b) The sample size is large enough since the smallest expected value is larger than 5.
c) The sample size is large enough since the smallest observed value is larger than 5.
Insert your answer (a, b, or c).