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QUAN 2010 Notes: Introduction to Business Statistics

Omar Ortega

Section 10.4 Hypothesis Testing for \(\mu\) When \(\sigma\) is Unknown

Up to this point, we’ve assumed that we know \(\sigma\text{,}\) the population mean. But as we saw in the last chapter on confidence intervals, if we don’t know \(\mu\text{,}\) it’s pretty unrealistic that we would know \(\sigma\text{.}\) So just like we did when creating confidence intervals, we will estimate \(\sigma\) with the sample standard deviation, \(s\text{.}\) This means that we will be dealing with the Student’s \(t\)-distribution in place of the normal distribution.
Hence, the test statistic for hypothesis tests of this type are computed with the formula:
\begin{equation*} t_{\overline{x}}=\frac{\overline{x}-\mu}{s/\sqrt{n}} \end{equation*}

Exercise 10.4.1.

(Donnelly 9.20)
In 2017, the average credit score for loans that were purchased by a government-sponsored mortgage loan company was 742. A random sample of 35 mortgages recently purchased by the company was selected, and it was found that the average credit score was 752 with a sample standard deviation of 22. Using \(\alpha = 0.05\text{,}\) is there enough evidence from this sample to conclude that the average credit score for mortgages purchased by the company has increased since 2017? Use the traditional method of hypothesis testing.
Answer.
\(n=35\rightarrow df=34,\;\; \overline{x}=752,\;\; s=22,\;\; \alpha=0.05\)
\(H_0: \; \mu\leq 742\)
\(H_1: \; \mu\gt 742 \;\; (\leftarrow \text{ right-tail test})\)
critical value: \(t_{.05}=T.INV(.95,34)\approx 1.691\)
test statistic:
\begin{equation*} t_{\overline{x}}=\frac{752-742}{22/\sqrt{35}}\approx 2.69 \end{equation*}
Since \(t_{\overline{x}}\gt t_{\alpha}\text{,}\) the test statistic falls in the rejection region; therefore, reject \(H_0\text{.}\)
There is enough evidence to conclude that the average credit score for mortgages has increased since 2017.
In our next example, we will use the p-value method of hypothesis testing. Unfortunately, tables for the Student’s \(t\)-distribution are limited in the precision they can provide for our p-values. Therefore, we will rely on Excewl to compute the p-values using the following formulas:
\begin{equation*} \boxed{T.DIST(x, df, cumulative)}\;\;\;\; \boxed{T.DIST.RT(x, df)}\;\;\;\; \boxed{T.DIST.2T(x, df)} \end{equation*}
where
\(x=\) the test statistic, \(t_{\overline{x}}\)
\(df=\) the degrees of freedom
cumulative \(=\) TRUE (since we want the accumulated area left of our test statistic)

Exercise 10.4.2.

(Donnelly 9.19)
According to the financial reports by Snapchat, the average daily user of Snapchat created 19 messages, or “snaps,” per day in Q3 2017. A college student wants to find out if the number of Snapchat messages has changed since Q3 2017 and creates a random sample using information from students on her campus for the current semester. The results are found in this lesson’s Excel file. Using \(\alpha = 0.01\text{,}\) test the hypothesis that the number of Snapchats sent by the average Snapchat user has not changed since Q3 2017. Use the p-value method of hypothesis testing.
Answer.
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