Paired
Example: A study is done where \(35\) overweight individuals are selected
for a new nutrition program for weight loss. The study tracks the
individuals weight at the beginning of the study and records their
weight again \(90\) days later. Below
is a table for their weight loss:
| 206.6 |
215.3 |
-8.7 |
| 211.5 |
213.3 |
-1.8 |
| 238.4 |
204.1 |
34.3 |
| 216.1 |
200.4 |
15.7 |
| \(\vdots\) |
\(\vdots\) |
\(\vdots\) |
| 228.2 |
218.8 |
9.4 |
From this table we find that the average weight loss is \(8.9\) lbs with a standard deviation of the
differences is \(20.49009\). Set up a
hypothesis test to determine whether, on average, the difference is real
or simply due to randomness.
\(H_0:\)
\(H_A:\)
Example: A local Davidson resident wants to test if the difference
in price for 15 staple groceries is significant between Harris Teeter
and Food Lion. After going to the stores they collect the below data for
15 items. At a significance level of \(.10\) conduct a hypothesis test to
determine if the difference is significant.:
- Some paired data has the sample statistics listed below. Use the
sample statistics to determine if there is a true difference between the
groups.
\(H_0:\)
\(H_A:\)
- Some paired data has the sample statistics listed below. Use the
sample statistics to determine if there is a true difference between the
groups.
\(H_0:\)
\(H_A:\)
- (This problem is a repeat from before, try to redo it using a t or z
- table instead of using R) A child is seeing how long they can hold
their breathe under water. The child thinks they can hold their breathe
for \(150\) seconds on average. The
child’s dad thinks it less than that. He samples his daughter holding
her breathe eight times and the results are \(144\), \(152\), \(138\), \(144\), \(136\), \(162\), \(158\), and \(142\). From the perspective of the dad,
perform a hypothesis test using a \(5
\%\) level of significance. Does the data provide sufficient
evidence to reject the null hypothesis?
\(H_0:\)
\(H_A:\)
- Pew Research asked a random sample of \(1000\) American adults whether they
supported the increased usage of coal to produce energy. Their sample
showed that \(46 \%\) of support
increased coal usage. Set up hypotheses to evaluate whether a majority
of American adults support or oppose the increased usage of coal.
\(H_0:\)
\(H_A:\)
- According to a report on sleep deprivation by the Centers for
Disease Control and Prevention, the proportion of California residents
who reported insufficient rest or sleep during each of the preceding
\(30\) days is \(8.0 \%\), while this proportion is \(8.8 \%\) for Oregon residents. These data
are based on simple random samples of \(11,545\) California and \(4,691\) Oregon residents. Make a hypothesis
test to determine if there is a real difference between the groups.
\(H_0:\)
\(H_A:\)
- According to a report on sleep deprivation by the Centers for
Disease Control and Prevention, the proportion of California residents
who reported insufficient rest or sleep during each of the preceding
\(30\) days is \(8.0 \%\), while this proportion is \(8.8 \%\) for Oregon residents. These data
are based on simple random samples of \(11,545\) California and \(4,691\) Oregon residents. Calculate a \(95 \%\) confidence interval for the
difference between the proportions of Californians and Oregonians who
are sleep deprived and interpret it in context of the data. (Pay
attention to how this compares to the hypothesis test approach)