**CppQuant Answer**

When constructing confidence intervals, the t-distribution is used when the *population* standard deviation is unknown.

- In Excel, m = AVERAGE(200, 240, 300, 410, 450, 600) = 366.6667
- sample standard deviation = variance^0.5, where variance = sum( (x(0)-m)^2 + ... + (x(n)-m)^2 )/n 60.9736
- In Excel, s = STDEV(200, 240, 300, 410, 450, 600) = 149.3541652

- Standard error of the mean = s / (N^0.5) = 149.3541652 / (6^0.5) = 60.97
- df = 6-1 = 5
- t = 2.571 (from the t-table)
- Confidence interval = Point Estimate +/- Margin of Error = Point Estimate +/- Reliability Factor * Standard Error
- Lower Limit = 366.67 - 2.571*60.87 = 209.904
- Upper limit = 366.67 + 2.571*60.87 = 523.430

- the researcher can be 95% sure that the mean reading speed of high-school graduates is between 209.904 and 523.430.

*Bonus Points*

- When use t-distribution to construct confidence intervals, one assumes
- normal distribution
- the population standard deviation is unknown.
- Scores are sampled randomly and are independent

- T-distributions are spread out MORE than a normal distribution with u = 0, s = 1.
- T-distributions are robust against nonnormality of the population except in the case of outliners or strong skewness.
- sample size < 15: can use T-distribution only if the data are close to normal.
- sample size >= 15: can use T-distribution except in the presence of outliners or strong skewness.
- sample size >= 40: can always use T-distribution.

- To shorten a given confidence interval, one should increase the sample size.

*Category: Quantitative Analysis*

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