Testing Claims about a Mean (Population Mean, $oldsymbol{ ext{μ}}$ Unknown)

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Testing Claims about a Mean (μ is Unknown)

Introduction

In inferential statistics, we often wish to test claims about a population mean. When the population standard deviation () is unknown, we use the t-distribution to conduct hypothesis tests. This section outlines the requirements, methods, and examples for testing claims about a mean when is unknown.

Requirements

Simple Random Samples: The sample data must be collected using a simple random sampling method.
Population Distribution: Either the population is normally distributed, or the sample size is sufficiently large ().

Notation

: Sample standard deviation
: Sample size
: Sample mean
: Population mean
: Null hypothesis
: Alternative hypothesis

Test Statistic

When is unknown, the test statistic for the mean is:

This statistic follows a t-distribution with degrees of freedom.

Properties of the t-Distribution

The t-distribution varies based on sample size (degrees of freedom).
It has the same general bell shape as the normal distribution.
The mean of the t-distribution is 0.
As sample size increases, the t-distribution approaches the normal distribution.

P-value Method

The p-value method is commonly used to determine whether to reject the null hypothesis. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the value observed under the null hypothesis.

Example 1

Scenario: Testing if the population mean of student course evaluations is equal to 4.00.
Sample Data: , ,
Hypotheses:
Test Statistic:
P-value:
Conclusion: Since , we fail to reject . There is not sufficient evidence to reject the claim that the population mean is 4.00.

Example 2

Scenario: Measure your pulse rate in a one-minute span. Assume pulse rates are normally distributed.
Task: Conduct a t-test to determine if your professor's resting pulse rate is different from the class mean, using a significance level of 0.05.

Critical Value Method

Alternatively, the critical value method can be used, but it requires calculation of critical t-values, which may not be readily available on calculators. This method is less commonly used in practice when p-values are easily accessible.

Confidence Interval Method

Confidence intervals can be used to test claims about a mean. If the claimed mean falls within the confidence interval, we fail to reject the null hypothesis.

Example 3

Scenario: Using the data from Example 1, construct a 95% confidence interval for the mean.
Calculation:
Conclusion: Since 4.00 is within the interval, we fail to reject the null hypothesis.

Example 4

Scenario: Students estimate the length of one minute without using a clock. Test if the mean is 60 seconds.
Sample Data:
Time (s)
69
39
65
62
64
63
66
48
64
70
96
91
45
21
60
63
Hypotheses:
Significance Level:
Confidence Interval:
Conclusion: Since 60 is within the interval, we fail to reject the null hypothesis. Students are reasonably good at estimating the length of one minute.

Time (s)
69	39	65	62	64	63	66	48
64	70	96	91	45	21	60	63

Summary Table: Methods for Testing Claims about a Mean

Method	Key Steps	Decision Rule
P-value Method	Calculate t-statistic, find p-value, compare to significance level	Reject if
Critical Value Method	Calculate t-statistic, compare to critical t-value	Reject if
Confidence Interval Method	Construct confidence interval for mean	Reject if claimed mean is outside interval

Additional info:

The t-distribution is robust to moderate departures from normality, especially for larger sample sizes ().
Always check assumptions before conducting hypothesis tests.

Testing Claims about a Mean (Population Mean, $oldsymbol{ ext{μ}}$ Unknown)