`[1] 1.644854`

STAT 120

Bastola

Mosquitoes are tested with two mouse groups: Malaria-infected (experimental) and Healthy (control)

Stages of malaria in mice:

`Stage 1`

: Non-infectious (Days 1-8)`Stage 2`

: Infectious (Days 9-28)`Response Variable`

: Whether the mosquito approaches a human.

`Research Questions`

: Do mosquitoes behave differently around malaria-infected versus healthy mice? Does the infection stage affect their behavior?

Malaria parasites would benefit if

- Mosquitoes approached humans
`less often`

after being exposed, but before becoming infectious, because humans are risky - Mosquitoes approached humans
`more often`

after becoming infectious, to pass on the infection

We’ll first look at the mosquitoes before they become infectious (days 1-8).

\(p_C:\) proportion of controls to approach human

\(p_E:\) proportion of exposed to approach human

What are the relevant hypotheses?

A. \(\mathrm{H}_0: p_{\mathrm{E}}=p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}<p_{\mathrm{C}}\)

в. \(\mathrm{H}_0: p_{\mathrm{E}}=p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}>p_{\mathrm{C}}\)

- \(\mathrm{H}_0: p_{\mathrm{E}}<p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}=p_{\mathrm{C}}\)

D. \(H_0: p_{\mathrm{E}}>p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}=p_{\mathrm{C}}\)

**p̂ _{E} - p̂_{C} = 20/113 - 36/117 = 0.177 - 0.308 = -0.131**

**What do you notice?**

For random samples with a sufficiently large sample size, the distribution of sample statistics for a mean or a proportion is normally distributed

The more skewed the original distribution of data/population is, the larger \(n\) has to be for the CLT to work

- For quantitative variables that are not very skewed, \(\boldsymbol{n} \geq \mathbf{3 0}\) is usually sufficient
- For categorical variables, counts of at least 10 within each category is usually sufficient

\[\begin{aligned}
A. \quad & \mathrm{N}(0,-0.131) \\
B. \quad& \mathrm{N}(0,0.056) \\
C. \quad& \mathrm{N}(-0.131,0.056) \\
D. \quad& \mathrm{N}(0.056,0)
\end{aligned}\]

Suppose: randomization distribution is bell shaped.

`Center`

: hypothesized null parameter value`Spread`

: the standard error given in the randomization graph (or by formula)`P-value`

: computed from the normal model the “usual” way - the chance of being as extreme, or more extreme, than the observed statistic.

The standardized test statistic (also known as a z-statistic) is \[\begin{align*} z=\frac{\text { statistic }-\text { null }}{S E} \end{align*}\]

Calculating the number of standard errors a statistic is from the null lets us assess extremity on a common scale.

Does infecting mosquitoes with Malaria actually impact the mosquitoes’ behavior to favor the parasite?

- After the parasite becomes infectious, do infected mosquitoes approach humans more often, so as to pass on the infection?

For the data after the mosquitoes become infectious (Days \(9-28\)), what are the relevant hypotheses?

\[\begin{align} \boldsymbol{p}_{C}: & \text{proportion of controls to approach human}\\ \boldsymbol{p}_{E}: & \text{proportion of exposed to approach human} \end{align}\]

\[\begin{aligned} A. \quad & \mathrm{H}_0: p_{\mathrm{E}}=p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}<p_{\mathrm{C}} \\ B. \quad & \mathrm{H}_0: p_{\mathrm{E}}=p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}>p_{\mathrm{C}} \\ C. \quad & \mathrm{H}_0: p_{\mathrm{E}}<p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}=p_{\mathrm{C}} \\ D. \quad & \mathrm{H}_0: p_{\mathrm{E}}>p_{\mathrm{C}}, \mathrm{H}_{\mathrm{a}}: p_{\mathrm{E}}=p_{\mathrm{C}} \end{aligned}\]
**p̂ _{E} - p̂_{C} = **

20/113 - 36/117 =

0.177 - 0.308 = -0.131

**p̂ _{E} - p̂_{C} = **

37/149 - 14/144 =

0.248 - 0.097 = 0.151

The difference in proportions is 0.151 and the standard error is 0.05. Is this significant?

A. Yes

B. No

It appears that mosquitoes infected by malaria parasites do, in fact, behave in ways advantageous to the parasites!

- Exposed mosquitos are
`less likely`

to approach before becoming infectious (so more likely to stay alive) - Exposed mosquitos are
`more likely`

to approach humans after becoming infectious (so more likely to pass on disease)

\[\begin{align*} z=\frac{\text { sample statistic }-\text { null value }}{\text { SE }} \end{align*}\]

**From original data**

**From H _{o}**

**From randomization distribution**

Suppose: bootstrap distribution is bell-shaped.

`Center`

: sample statistic`Spread`

: the standard error given in the bootstrap graph (or by formula)

If a bootstrap distribution is normally distributed, we can write it as

A. \(\mathrm{N} (parameter, SD)\)

B. \(\mathrm{N} (statistic, SD)\)

C. \(\mathrm{N} (parameter, SE)\)

D. \(\mathrm{N} (statistic, SE)\)

To get a \(95 \%\) confidence interval we compute: \[statistic \pm 2(SE)\]

Why 2 SE’s?

- \(95 \%\) of all sample means fall within 2 SE’s of the population mean
- The value 2 is a z-score!
- Well, actually the precise z-score under a normal model is \(z=1.96\) instead of 2 !

**95% of all values fall within 1.96 SE’s of the mean**

If a statistic is normally distributed, we find a confidence interval for the parameter using \[statistic \pm z^* SE\] where the area between \(-z^*\) and \(+z^*\) in the standard normal distribution is the desired level of confidence.

- Please download the Class-Activity-16 template from moodle and go to class helper web page

`30:00`