t test for one mean using Microsoft Excel

Reproduced with permission from
Gerry's Handbook of Statistical Analysis for Non-math Types.

We use this test when:

1. The scale of the data is interval or ratio.
2. We have to assume the data follows a normal distribution since we don't have enough history to determine if the population follows a normal distribution.
3. Similarly, we don't know the population standard deviation for the same reason as above.
4. The sample size is under 30 (so we can't use the Central Limit Theorem).

Example: A new restaurant's business plan projected that the average bill would be more than $30. After the first day's operations, the owners sampled 12 bills. These were the results:

Are the business plan projections accurate? Test at a 5% level of significance.

This is a right-tail test because we want to see if the average bill is more than $30.

From Gerry's Stats Tools, we will run through "Which test do I choose?" to see why we will use the t test for one mean.

Question 1 - Choose which type of test.
We choose means.

Question 2 - How many means are we testing?
We choose 1 mean.

Question 3 - What scale is the data?
Since we are dealing with money, the scale is ratio. We choose Interval/ratio.

Question 4 - What is the sample size?
Since the sample size is 12, we choose Under 30.

Question 5 - Is the data normal?
We don't know that, do we? So, we click the "I don't know" button which brings up the normality test:

• The input range is a1:a12.
• We choose b1 for the output range.

Here is the output:

The data is normally distributed
at a 5% level of significance
Critical value: 0.242
Test statistic: 0.139

Oh good, the data is normal.

Question 5 - Is the data normal?
Yes.

Question 6 - Is the population standard deviation known?
No.

The last screen tells us that we use the t test for one mean. To proceed to the actual test, we click the "Go to the test" button. From there:

• The input range is a1:a12.
• We leave the population size box blank.
• For the population standard deviation, we leave the box blank.
• We set the level of significance at 0.05.
• We choose right tail for the type of test.
• The hypothesized mean is 30.
• We choose b1 for the output range.

This is the output:

t test for one mean
Ho: population mean is not greater than 30
Ha: population mean is greater than 30
Reject Ho if test statistic > 1.796
Test statistic = 1.511
P-value = 0.079
Do not reject Ho
Conclude population mean is not greater than 30
95% confidence interval: 29.035 to 35.195

Since this is a right-tail test, we reject the null hypothesis if the test statistic is greater than 1.796. Since the test statistic is 1.511, we do not reject the null hypothesis. We conclude the population mean is not greater than $30. In the context of the problem, we conclude that the business plan projections are not accurate.

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