```
** Bootstrap interval for mean
Observed X : 20.66667
Mean of bootstrap distribution: 20.66403
Standard error of bootstrap distribution: 1.09819
Bootstrap percentile interval
2.5% 97.5%
18.50000 22.66667
*--------------*
```

STAT 120

Bastola

The higher the standard error of a statistic, the \(\ldots \ldots\) the uncertainty surrounding the statistic.

- higher
- lower

- Point estimates are almost always not accurate
- Uncertainty in point estimates measured by the Standard Error (SE)
- A plausible range of values for the population parameter is more reliable
- Interval Estimate: An interval estimate is an interval of numbers within which the parameter value is believed to fall

How accurate is an estimate of \(60\%\)?

Link to the Gallup poll

- Interval estimate: \(60\% \pm 3\% = (57\% , 63\%)\)
- The percentage of American adults who would vote for a Muslim for president is likely between \(57\%\) and \(63\%\).

The margin of error measures how accurate a point estimate is likely to be in estimating a parameter.

To determine the margin of error, we can use the statistic’s sampling distribution and standard error

- A confidence interval is an interval containing the most believable values for a parameter
- A confidence interval is centered on the point estimate and extends a certain number of standard errors on either side of the estimate
- The confidence level tells us what percent of the intervals will contain the population parameter.
- A 95% confidence interval means that if we were to draw numerous samples and calculate their confidence intervals, about 95% of these intervals would be expected to contain the true population parameter.

- always about the population
- not probability statements
- only about population parameters, not individual observations
- only reliable if the sample statistic they’re based on is an unbiased estimator of the population parameter

If the sampling distribution is relatively symmetric and bell-shaped, a \(95\%\) confidence interval can be estimated using \[\mathbf{statistic} \pm 2 \times \mathbf{SE}\]

Let’s all go to Statkey web app.

- The parameter is fixed
- The statistic is random (depends on the sample)
- The interval is also random (depends on the statistic)
- Confidence level is the proportion of intervals that capture the true parameter

- Repeated sampling is needed to compute the standard error of a sample statistic
- Can estimate the SE from a bootstrap distribution
- Use this SE to compute a confidence interval for an unknown parameter

A bootstrap distribution is the distribution of many bootstrap statistics.

- The standard deviation of this distribution is called the bootstrap standard error of the statistic.
- The bootstrap distribution is centered near the original sample mean.

```
** Bootstrap interval for mean
Observed X : 20.66667
Mean of bootstrap distribution: 20.66403
Standard error of bootstrap distribution: 1.09819
Bootstrap percentile interval
2.5% 97.5%
18.50000 22.66667
*--------------*
```

- Please download the Class-Activity-8 template from moodle and go to class helper web page

`30:00`