Question

Q: Would like to understand and solve this example from Probability and Statistical Inference with the steps of the solution to better understand, thanks.

**Please give the step by step with details to completely see how the solution came about, plenty of thanks.

1) Two detergents were tested for their ability to remove stains of a certain type. Any inspector judge the first one to be successful on 63 out of 93 independent trials and the second one to be successful on 42 out of 78 independent trials. The respective frequencies of success are 0.692 and 0.532.

(a) Find a 90% confidence interval for the population proportion difference.

Answer 1

Solution: This problem can be solved using Hypothesis Test: Difference between proportion.

The test procedure , called two proportion test.

This approach consists of four steps: 1) state the hypothesis 2) formulate an analysis plan 3) analyze sample data and 4) interpret result.

Every hypothesis test requires the analyst to state null hypothesis and alternative hypothesis

  

d is the difference between two population proportion

Here, Two detergents were tested for their ability to remove stains of certain type

The test is to find out whether their is no difference in their ability to remove stain or there exist some difference in their ability

For the first detergent sample size 93 randomly selected from population and out of which 63 cases it is found to be successful

For the second type of detergent sample size 42 randomly selected from population and out of which 78 cases it is found to be successful

Respective proportion of success of first and second detergents population are

When the null hypothesis states that there is no difference between the two population proportions (i.e d=P₁-P₂=0)

the null and alternative hypothesis for a two-tailed test are often stated in the following form.

Formulate an Analysis Plan:

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use the two-proportion z-test to determine whether the hypothesized difference between population proportions differs significantly from the observed sample difference.

Lets define 0.10 as significance level in our test of proportion of two type of detergent and use the two proportion z test

Analyze Sample Data:

Using sample data, complete the following computations to find the test statistic and its associated P-Value.

Pooled sample proportion. Since the null hypothesis states that P₁=P₂, we use a pooled sample proportion (p) to compute the standard error of the sampling distribution.

In our test

Standard error. Compute the standard error (SE) of the sampling distribution difference between two proportions

In our test:

Test statistic. The test statistic is a z-score (z) defined by the following equation.

In our test

P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a z-score. we can find p-value in z table for 0.10 singificance level in z table against this z score or test statistic.It is a two tailed test

The p-value  is .061484.

Interpret Results:

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

In out test p-value is 0.061484 <0.10

Hence, result is significance

We can conclude that there is a difference of ability exist between the two type of detergent for removing stain .

90% confidence interval for the population proportion difference is given by

here

please put the value and calculate the confidence interval i am running out of time for any query comment me