Question

Problem #2

Bumble likes going to his local Subway and always orders a foot-long meatball sandwich there, as he finds the satisfaction he gets from a foot-long sandwich to be priceless. Over the past month, however, Bumble has been constantly hungry. He suspects the meatball sandwiches at his local Subway to be shrinking on average. Bumble has randomly sampled 20 of those ‘supposedly’ one-foot-long (12 inches) meatball sandwiches at that local Subway and has found a sample mean length of 11.8 inches with a sample standard deviation of 0.30 but does not know what to conclude from there. Please answer the following questions to help Bumble conduct an appropriate test to see if the mean length is shrinking at the 0.01 level of significance.

2-1 State the appropriate H₀ and H₁ for this test.

Separately fill in the parameter, equality, and value by reporting letters respectively from the following list to complete each hypothesis.

H₀:  - select -abcdef  - select -abcdef  - select -abcdef

H₁:  - select -abcdef  - select -abcdef  - select -abcdef

parameter              equality           value

1. µ                      a.  =                 a. 12
2. x̅                      b.  ≠                 b. 11.8
3. σ                      c. ≥                 c. 20
4. π                      d.  ≤                 d. 0.30
5. p̅                      e.  >                 e. 0.01
6. α                      f.  <                 f. 0

  

2-2 What would be the most appropriate critical value (i.e., z_crit or t_crit) for this test?

Separately fill in the name, sign, and value by reporting letters respectively from the following list to complete the critical value (e.g., to together state z_crit = +1.96).

- select -abcdef =  - select -abcdef  - select -abcdef

name:                    sign:                value:

1. z_crit                   a.  +                 a. 2.861
2. t_crit                   b.  −                 b. 2.845

c.  ±                 c. 2.539

                                                      d. 2.528

                                                      e. 1.96

                                                      f. 1.645

  

2-3 What is the p-value for this test?

Select from the list below and report the letter.  - select -abcdef

1. less than 0.005
2. between 0.005 and 0.01
3. between 0.01 and 0.025
4. between 0.025 and 0.05
5. between 0.05 and 0.1
6. greater than 0.1

2-4 Bumble thinks the p-value is the probability that the sandwiches are shrinking in size. Is this true? Please correct him if needed and interpret the p-value by directly explaining what this probability itself means.

Select the most appropriate statement from the list below and report the letter.  - select -abcdef

1. Yes, Bumble is correct. The p-value represents the probability that the alternative hypothesis is true.
2. No, Bumble is wrong. The p-value represents the probability that the sandwiches are NOT shrinking in size; so the lower p-value suggests that the sandwiches are indeed shrinking.
3. No, Bumble is wrong. The p-value indicates the probability of a sample mean being 11.8 inches or less if the true average length is really 12 inches.
4. No, Bumble is wrong. The p-value indicates the probability of the population mean being 12 inches if we observed a sample mean of 11.8 inches.
5. Yes, Bumble is correct. The p-value represents the probability that the null hypothesis is true and that he is not committing the Type I (α) error.
6. None of the above.

2-5 What is the most appropriate conclusion for this hypothesis test?

Select the most appropriate statement from the list below and report the letter.  - select -abcdef

1. Fail to reject H₀ and conclude that the sandwiches are not 12 inches on average.
2. Reject H₀ and conclude that the sandwiches are bigger than 12 inches on average.
3. Fail to reject H₀ and conclude that the sandwiches are 12 inches or bigger on average.
4. Reject H₀ and conclude that the sandwiches are smaller than 12 inches on average.
5. Fail to reject H₀ and conclude that the sandwiches are 12 inches or smaller on average.
6. Reject H₀ and conclude that the sandwiches are 12 inches on average.

Answer 1

2-1:

2-2

2-3:

From t table, for df =19 and t = -2.981 the p-values is less than less than 0.005.

2-4

Correct option :

No, Bumble is wrong. The p-value indicates the probability of a sample mean being 11.8 inches or less if the true average length is really 12 inches.

2-5:

Correct option:

Reject H0 and conclude that the sandwiches are smaller than 12 inches on average.