An article claimed that the typical supermarket trip takes a mean of 22 minutes. suppose that in an effort to test this claim, you select a sample of 50 shoppers at a local supermarket. the mean shopping time for the sample of 50 shoppers is 25.36 min. with a standard deviation of 7.24 minutes. using a 0.10 level of significance is there evidence that the mean shopping time at the local supermarket is different from the claimed value of 22 minutes?
***if i plug everything into Z= mean-22minutes/ standard deviation over sqrt of n is that the answer? i am soo confuseddd
Let me give you a little background and maybe this will help you understand a few of the concepts of hypothesis testing.
We use samples to support some hypothesis about a population. We make inferences about the population from sample data. When setting up a statistical test, we form two hypotheses: a null hypothesis and an alternative hypothesis. The null hypothesis tests the probability that the sample mean is very likely drawn from the population mean. If there is a low probability that the sample mean has been drawn from the population mean, we will reject the null hypothesis and accept the alternative hypothesis. The alpha level (also called the significance level) is the probability at which we reject the null and accept the alternative hypothesis. For example, if the alpha level is set at 0.05, there is a 5% probability that we will reject the null hypothesis. This is a very common alpha level used in statistics. Often we use tables to look for values that correspond to the alpha level stated. These values are known as critical or cutoff values. They cut off the distribution at that point. If we have an observed value (calculated from a formula) that exceeds our critical value or cutoff value from a table, then we have to reject the null hypothesis and accept the alternative hypothesis. If the observed value does not exceed the critical or cutoff value from a table, then we fail to reject the null hypothesis. How do we translate 0.05 into a critical value? It depends on the type of test we are doing. For a one-tailed test, we don't split the value. For a two-tailed test, we split the .05 into .025 and .025 for both tails of the distribution curve (a two-tailed test is like a confidence interval in that respect). If we use a z-table for a one-tailed test at .05 level of significance (meaning the alternative hypothesis is showing a specific direction like "less than" or "greater than" in its statement), then we will have a critical or cutoff value of z = -1.645 or it could be z = 1.645, depending on the direction. This will determine where we "draw the line" to reject the null hypothesis. If we use a z-table for a two-tailed test at .05 level of significance (meaning the alternative hypothesis is showing no specific direction and uses "does not equal" in its statement), then we will have a critical or cutoff value of z = + or - 1.96 (meaning either tail of the distribution curve). If we had a test result of z = +2.00 (for example), then we would have exceeded the positive critical value of +1.96 for this particular test result and the null hypothesis would be rejected in favor of the alternative hypothesis.
Now let's look at your problem.
You need to set up hypotheses, calculate the z-test statistic (since this is a z-test), then compare to the critical value from a z-table to determine whether or not to reject the null hypothesis.
Hypotheses:
Ho: µ = 22 -->this is the null hypothesis.
Ha: µ does not equal 22 -->this is the alternate or alternative hypothesis.
This would be a two-tailed or nondirectional test because the alternative hypothesis doesn't specify a specific direction.
The reason we know this is a two-tailed test is because the problem asks if there is evidence that the mean time is different, which means the results could be in either tail of the distribution.
Therefore, using a z-test formula:
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
With your data:
z = (25.36 - 22)/(7.24/√50)
Finish the calculation.
Now you will need to find the critical value at 0.10 level of significance using a z-table. Since this is a two-tailed test, we split the 0.10 into 0.05 and 0.05 for both tails of the distribution curve. Find the critical value, then compare to your test value. Determine whether or not to reject the null.
I hope this explanation will help.
To determine if there is evidence that the mean shopping time at the local supermarket is different from the claimed value of 22 minutes, you can conduct a hypothesis test.
Here's how you can go about it:
Step 1: State the hypotheses:
- Null hypothesis (H0): The mean shopping time is equal to 22 minutes.
- Alternative hypothesis (Ha): The mean shopping time is different from 22 minutes.
Step 2: Select the level of significance:
The level of significance, also known as alpha (α), is the probability of rejecting the null hypothesis when it is true. In this case, the level of significance is given as 0.10 (or 10%).
Step 3: Identify the appropriate test statistic:
Since you know the mean (25.36 minutes) and standard deviation (7.24 minutes) of the sample, and the sample size (n = 50) is large, you can use a Z-test for this hypothesis test.
Step 4: Formulate the test statistic:
The test statistic is calculated by subtracting the claimed value from the sample mean, and then dividing it by the standard deviation divided by the square root of the sample size.
Z = (sample mean - claimed value) / (standard deviation / sqrt(sample size))
= (25.36 - 22) / (7.24 / sqrt(50))
Calculating this formula will give you the value of Z.
Step 5: Determine the critical value(s):
The critical value(s) will help you decide whether to reject or fail to reject the null hypothesis. Since the alternative hypothesis is two-sided (mean is different from 22 minutes), you need to find the critical values for a two-tailed test.
To find the critical value(s), you can use a Z-table and look up the value that corresponds to your level of significance (α/2 = 0.10/2 = 0.05) for both tails. The critical values will be negative and positive, denoted as -Zα/2 and +Zα/2, respectively.
Step 6: Make a decision:
Compare the calculated test statistic (Z) from Step 4 with the critical value(s) from Step 5.
If the test statistic falls in the rejection region, which is when it is smaller than -Zα/2 or larger than +Zα/2, you will reject the null hypothesis.
If the test statistic does not fall in the rejection region, you will fail to reject the null hypothesis.
Step 7: Interpret the decision:
Based on the decision in Step 6, you can interpret whether there is evidence to support the claim that the mean shopping time is different from 22 minutes.
If you reject the null hypothesis, it suggests that there is evidence to support the claim that the mean shopping time is different from 22 minutes.
If you fail to reject the null hypothesis, it implies that there is not enough evidence to suggest a difference in the mean shopping time from 22 minutes.
Remember to calculate the actual values and compare the test statistic with the critical value(s) to make a final decision based on your findings.