Stat120 Spring24 - Additional topics in testing

Question 1

A new blood thinning drug is being tested against the current drug in a double-blind experiment. Is there evidence that the mean blood thinness rating is higher for the new drug? Using \(n\) for the new drug and \(o\) for the old drug, which of the following are the null and alternative hypotheses?

A. \(\mathrm{H}_0: \mu_{\mathrm{n}}>\mu_0 \quad\mathrm{Vs}\quad \mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}}=\mu_{\mathrm{o}}\)

B. \(\mathrm{H}_0: \mu_{\mathrm{n}}=\mu_0\quad\mathrm{Vs}\quad\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}} \neq \mu_0\)

C. \(\mathrm{H}_0: \mu_{\mathrm{n}}=\mu_0\quad\mathrm{Vs}\quad\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}}>\mu_0\)

D. \(\mathrm{H}_0: \bar{x}_n=\bar{x}_o \quad \mathrm{Vs} \quad \mathrm{H}_{\mathrm{a}}: \bar{x}_n \neq \bar{x}_o\)

E. \(\mathrm{H}_0: \bar{x}_n=\bar{x}_o\quad\mathrm{Vs}\quad\mathrm{H}_{\mathrm{a}}: \bar{x}_n>\bar{x}_o\)

Click for answer

The correct answer is C.

Question 2

A new blood thinning drug is being tested against the current drug in a double-blind experiment, and the hypotheses are: \[\mathrm{H}_0: \mu_{\mathrm{n}}=\mu_{\mathrm{o}} \quad \text { vS } \quad \mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}}>\mu_{\mathrm{o}}\] What does a Type I Error mean in this situation?

A. We reject \(\mathrm{H}_0\)

B. We do not reject \(\mathrm{H}_0\)

C. We find evidence the new drug is better when it is really not better.

D. We are not able to conclude that the new drug is better even though it really is.

E. We are able to conclude that the new drug is better.

Click for answer

The correct answer is C.

Question 3

A new blood thinning drug is being tested against the current drug in a double-blind experiment, and the hypotheses are: \[\mathrm{H}_0: \mu_{\mathrm{n}}=\mu_{\mathrm{o}} \quad \text { vS } \quad \mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}}>\mu_{\mathrm{o}}\]

What does a Type II Error mean in this situation?

A. We reject \(\mathrm{H}_0\)

B. We do not reject \(\mathrm{H}_0\)

C. We find evidence the new drug is better when it is really not better.

D. We are not able to conclude that the new drug is better even though it really is.

E. We are able to conclude that the new drug is better.

Click for answer

The correct answer is D.

Question 4

A new blood thinning drug is being tested against the current drug in a double-blind experiment, and the hypotheses are: \[\mathrm{H}_0: \mu_{\mathrm{n}}=\mu_{\mathrm{o}} \quad \text { vS } \quad \mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{n}}>\mu_{\mathrm{o}}\]

If the new drug has potentially serious side effects, we should pick a significance level that is:

A. Relatively small (such as \(1 \%\))

B. Middle of the road \((5\%)\)

C. Relatively large (such as \(10 \%\))

Click for answer

The correct answer is A. Since a Type 1 error would be serious, we use a small significance level

Factors Affecting Type I and Type II Error Rates

Sample Size

Larger Sample Sizes:
- Improve the accuracy of the test.
- Decrease the chance of Type II error.
Smaller Sample Sizes:
- Increase the risk of not detecting a true effect (Type II error).

Factors Affecting Type I and Type II Error Rates

Variability

Greater Variability:
- Increases chances of both Type I and Type II errors.
Lesser Variability:
- Reduces chances of both errors.

Note: Ensuring a sufficiently large sample and considering the inherent variability in the population are vital in hypothesis testing.

Randomization distribution with \(n=10\)

Randomization distribution with \(n=100\)

Bootstrap Vs Randomization

Bootstrap Distribution

Our best guess at the distribution of sample statistics
Centered around the observed sample statistic
Simulate sampling from the population by resampling from the original sample

Randomization Distribution

Our best guess at the distribution of sample statistics, if \(\mathrm{H}_0\) were true
Centered around the null hypothesized value
Simulate samples assuming \(\mathrm{H}_0\) were true

Big difference: a randomization distribution assumes \(H_0\) is true, while a bootstrap distribution does not

bootstrap or randomization?

Using confidence intervals for hypothesis testing

If a \(95 \%\) CI contains the parameter in \(\mathrm{H}_0\), then a two-tailed test should not reject \(\mathrm{H}_0\) at a \(5 \%\) significance level.

If a \(95 \%\) CI misses the parameter in \(\mathrm{H}_0\), then a two-tailed test should reject \(\mathrm{H}_0\) at a \(5 \%\) significance level.