AP Statistics Curriculum 2007 Hypothesis L Mean

General Advance-Placement (AP) Statistics Curriculum - Testing a Claim about a Mean: Large Samples

We already saw how to construct point and interval estimates for the population mean in the large sample case. Now, we show how to do hypothesis testing about the mean for large sample-sizes.

Background

• Recall that the population mean may be estimated by the sample average, \(\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}\), of random sample {\(X_1, X_2, X_3, \cdots , X_n\)} of the procees.

• For a given small \(\alpha\) (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the \((1-\alpha)100%\) Confidence interval for the mean is constructed by

\[CI(\alpha): \overline{x} \pm z_{\alpha\over 2} E,\]

where the margin of error E is defined as

\[E = \begin{cases}{\sigma\over\sqrt{n}},& \texttt{for-known}-\sigma,\\ {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}},& \texttt{for-unknown}-\sigma.\end{cases}\]

and \(z_{\alpha\over 2}\) is the critical value for a Standard Normal distribution at \({\alpha\over 2}\).

Hypothesis Testing about a Mean: Large Samples

• Null Hypothesis\[H_o: \mu=\mu_o\] (e.g., 0)
• Alternative Research Hypotheses:
  • One sided (uni-directional)\[H_1: \mu >\mu_o\], or \(H_o: \mu<\mu_o\)
  • Double sided\[H_1: \mu \not= \mu_o\]

Known Variance

• Test statistics:

\[Z_o = {\overline{x} - \mu_o \over \sigma} \sim N(0,1)\].

Unknown Variance

• Test statistics:

\[T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim N(0,1)\].

Example

Let's revisit the number of sentences per advertisement example, where we measure of readability for magazine advertisements. A random sample of the number of sentences found in 30 magazine advertisements is listed below. Suppose we want to test a null hypothesis\[H_o: \mu=20\] against a double-sided research alternative hypothesis\[H_1: \mu \not= 20\].

16

9

14

11

17

12

99

18

13

12

5

9

17

6

11

17

18

20

6

14

7

11

12

5

18

6

4

13

11

12

We had the following 2 sample statistics computed earlier \[\overline{x}=\hat{\mu}=14.77\] \[s=\hat{\sigma}=16.54\]

As the population variance is not given, we have to use the T-statistics \[T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})} \sim T(df=29)\].

Hands-on activities

• See the SOCR Confidence Interval Experiment.
• Sample statistics, like the sample-mean and the sample-variance, may be easily obtained using SOCR Charts. The images below illustrate this functionality (based on the Bar-Chart and Index-Chart) using the 30 observations of the number of sentences per advertisement, reported above.

---

References

---

• SOCR Home page: http://www.socr.ucla.edu

---

---

Translate this page:

(default)	Deutsch	Español	Français	Italiano	Português	日本語	България	الامارات العربية المتحدة	Suomi	इस भाषा में	Norge
한국어	中文	繁体中文	Русский	Nederlands	Ελληνικά	Hrvatska	Česká republika	Danmark	Polska	România	Sverige