The Five Steps in The Hypothesis Testing Procedure

The Five Steps in The Hypothesis Testing Procedure are discussed below while solving the question.

State the null hypothesis and the alternate hypothesis. 

The null hypothesis to be tested is: “There are no statistically significant differences of the six measurements presenting sufficient evidence to indicate that the average weight of the diamonds produced by the new process is more than 0.5 karat .”

The alternate hypothesis to be tested is: “There are a statistically significant difference of the six measurements presenting sufficient evidence to indicate that the average weight of the diamonds produced by the new process is more than 0.5 karat .”

Selection of the appropriate test statistic and level of significance. 

Typically , when testing a hypothesis of a proportion, the z-statistic or z-test, and the formula is preferred.

Again, the z-statistic or the t-statistic is used when testing a hypothesis of a mean but must meet the following conditions;

The standard deviation, σ, of the population must be known, and the data is normally distributed or the sample size n > 30, to use the normal distribution (z-statistic). On the other hand, the standard deviation, σ, of the population is not known, and the data is normally distributed or the sample size n > 30, then t distribution (t-statistic) is used.

To choose the level of significance, a traditional guideline: (a) the 0.10 level for political polling, (b) the 0.05 level for consumer research projects, and (c) the 0.01 level for quality assurance work is followed .

Stating the decision rules. 

The decision rules state the conditions under which the null hypothesis will be accepted or rejected. The level of significance determines the critical value for the test-statistic . The significant value is the value that divides the non-reject region from the reject region. Here, involve deciding on which Confidence interval to use; 95% or 99% which are common. In this case, I have decided to use the 95% confidence interval.

Computing the appropriate test statistic and make the decision. In this task, the z-statistic has been used and involve the following processes; Step 1 

This is involving identification of the number of observations; n after that calculating its mean score; X-bar and finally, the standard deviation is also calculated ; s. 

In Option 1;

Number of observations; n = 6

The mean is given by;

The standard deviation; s is calculated as follows;

The next is to apply the Z formula which is;

= 0.00011232

Compare the computed test statistic with critical value. If the computed value is within the rejection region(s), we reject the null hypothesis; otherwise, we do not reject the null hypothesis.

Here are the already known Z values for some of the common Confidence intervals;

Since the computed value is not within the rejection region(s), I reject the null hypothesis.

Interpreting the decision. 

Based on the decision in Step 4, there are a statistically significant difference of the six measurements presenting sufficient evidence to indicate that the average weight of the diamonds produced by the new process is more than 0.5 karat .