Question

The Crown Bottling Company has just installed a new bottling process that will fill 16-ounce bottles of the popular Crown Classic Cola soft drink. Both overfilling and underfilling bottles are undesirable: Underfilling leads to customer complaints and overfilling costs the company considerable money. In order to verify that the filler is set up correctly, the company wishes to see whether the mean bottle fill, μ, is close to the target fill of 16 ounces. To this end, a random sample of 38 filled bottles is selected from the output of a test filler run. If the sample results cast a substantial amount of doubt on the hypothesis that the mean bottle fill is the desired 16 ounces, then the filler’s initial setup will be readjusted.

(a) The bottling company wants to set up a hypothesis test so that the filler will be readjusted if the null hypothesis is rejected. Set up the null and alternative hypotheses for this hypothesis test.

H₀ : μ (Click to select)=≠ 16 versus H_a : μ (Click to select)=≠ 16

(b) Suppose that Crown Bottling Company decides to use a level of significance of α = 0.01, and suppose a random sample of 38 bottle fills is obtained from a test run of the filler. For each of the following four sample means— x¯x¯ = 16.05, x¯x¯ = 15.97, x¯x¯ = 16.06, and x¯x¯ = 15.96 — determine whether the filler’s initial setup should be readjusted. In each case, use a critical value, a p-value, and a confidence interval. Assume that σ equals .1. (Round your z to 2 decimal places and p-value to 4 decimal places and CI to 3 decimal places.)

x¯x¯ = 16.05

z
p-value

CI            [, ]   (Click to select)Do not readjustReadjust

x¯x¯⁢ = 15.97

z
p-value

CI              [, ]   (Click to select)ReadjustDo not readjust

x¯x¯⁢ = 16.06

z
p-value

CI                  [, ]   (Click to select)ReadjustDo not readjust

x¯x¯⁢ = 15.96

z
p-value

CI              [, ] (Click to select)

i need help with this question.

Answer 1

(a) The null and alternative hypotheses for this test :

H₀ : μ =16 versus H_a : μ ≠ 16

(b) Given,

Sample size, n =38

Significance level , α = 0.01

Standard deviation , σ = 1

• For = 16.05

The z-statistic is computed as follows:

The p-value corresponding to z = 0.308, using standard normal table :

P-value =  0.7580

The critical value for α=0.01 is z_c​=2.576. The corresponding 99% confidence interval for the population mean μ is computed as shown below:

• For = 15.97

The z-statistic is computed as follows:

The p-value corresponding to z =- 0.185 ,using standard normal table :

P-value =  0.8533

The critical value for α=0.01 is z_c​=2.576. The corresponding 99% confidence interval for the population mean μ is computed as shown below:

• For = 16.06

The z-statistic is computed as follows:

The p-value corresponding to z = 0.37 ,using standard normal table :

P-value = 0.7115

The critical value for α=0.01 is z_c​= 2.576. The corresponding 99% confidence interval for the population mean μ is computed as shown below:

• For = 15.96

The z-statistic is computed as follows:

The p-value corresponding to z = - 0.2247 ,using standard normal table :

P-value =  0.8052

The critical value for α=0.01 is z_c​=2.576. The corresponding 99% confidence interval for the population mean μ is computed as shown below: