. Hypothesis Testing
o Please complete the following 5-step hypothesis testing procedure in your Learning Teams: A simple random sample of forty 20-oz bottles of Coke® from a normal distribution is obtained and each bottle is measured for number of ounces of Coke® in the bottle. The sample mean is 19.4 oz with standard deviation of 1.8 oz. Test the claim at an alpha significance level of 5% that on average there is exactly 19.8 oz of Coke® in each bottle.
2. Hypothesis Testing
o Please complete the following 5-step hypothesis testing procedure in your Learning Teams: A simple random sample of ten 20-oz bottles of Coke® from a normal distribution is obtained and each bottle is measured for number of ounces of Coke® in the bottle. The sample mean is 19.4 oz with standard deviation of 1.8 oz. Test the claim at an alpha significance level of 5% that on average there is more than 19.8 oz of Coke® in each bottle.
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Find an example of application of Normal Distribution (or approximately Normal Distribution) in your workplace or business (or any other business that you are familiar with). Prove that the variable has the characteristics of a Normal Distribution. Recall that the variable must be continuous and the distribution must be symmetrical (or approximately symmetrical). For a distribution to be approximately Normal, the values of Mean, Median, and Mode must be fairly close.
1. The Myers Summer Casual Store tells its customers that a special order will take six weeks (42 days). During the recent months, the owner has received several complaints that the special orders are taking longer than 42 days. A sample of 12 special orders delivered in the last month showed that the mean waiting time was 51 days, with a standard deviation of eight days. At the .01 level of significance, are customers waiting an average of more than 42 days?
To conduct hypothesis testing, we follow a 5-step procedure. Let's apply this procedure to both scenarios:
Scenario 1:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
Null hypothesis (H0): The average number of ounces of Coke in each bottle is equal to 19.8 oz.
Alternative hypothesis (Ha): The average number of ounces of Coke in each bottle is not equal to 19.8 oz.
Step 2: Choose the significance level (alpha):
The significance level (alpha) is given as 5% or 0.05.
Step 3: Collect and analyze the data:
A simple random sample of forty 20-oz bottles of Coke was obtained. The sample mean is 19.4 oz with a standard deviation of 1.8 oz.
Step 4: Calculate the test statistic and p-value:
For a sample mean with known standard deviation and a null hypothesis of equal means, we use a z-test. The test statistic can be calculated using the formula:
z = (x̄ - μ) / (σ / √n),
where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Using the given values: x̄ = 19.4, μ = 19.8, σ = 1.8, n = 40,
we can calculate the test statistic:
z = (19.4 - 19.8) / (1.8 / √40)
Step 5: Make a decision and interpret the results:
Using a z-table or statistical software, we can find the critical z-value(s) corresponding to the chosen significance level (alpha) and the type of alternative hypothesis (two-tailed, one-tailed greater, or one-tailed less). In this case, since the alternative hypothesis is two-tailed, we could find the two critical z-values.
Next, we compare the calculated test statistic to the critical z-values to make a decision. If the calculated test statistic falls within the critical region (outside the range of critical z-values), we reject the null hypothesis. Otherwise, if the calculated test statistic falls within the non-critical region (inside the range of critical z-values), we fail to reject the null hypothesis.
Finally, we interpret the results and make conclusions based on the decision made in step 5.
Scenario 2:
In this scenario, the claim is that on average, there is more than 19.8 oz of Coke in each bottle. Therefore, the null and alternative hypotheses can be stated as follows:
Null hypothesis (H0): The average number of ounces of Coke in each bottle is equal to 19.8 oz.
Alternative hypothesis (Ha): The average number of ounces of Coke in each bottle is greater than 19.8 oz.
Similarly, by following the remaining steps of the 5-step hypothesis testing procedure, we can calculate the test statistic, compare it to the critical value(s), and make a decision. The interpretation and conclusion will be based on the decision made in step 5.
Remember, in hypothesis testing, it is important to define the null and alternative hypotheses correctly, choose an appropriate significance level, analyze the data accurately, calculate the test statistic correctly, and interpret the results appropriately.