## STA286_Assignment_Report.pdf-pg. 12 Appendix C: Approximation toshowing page 13-14 out of 23

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pg. 12

Appendix C: Approximation to Normal Distribution

Figure 14: Normal Quantile-Quantile plot for

Typing Time on Laptop Keyboard

Figure 15: Normal Quantile-Quantile plot for

Number of Errors on Laptop Keyboard

Figure 16: Normal Quantile-Quantile plot for

Typing Time on Nexus Keyboard

Figure 17: Normal Quantile-Quantile plot for

Number of Errors on Nexus Keyboard

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pg. 13

The above quantile-quantile plots show an almost linear relationship between the

standard normal quantiles and sample quantiles. This suggests that the data can also be

approximated as a normal distribution. Although the data fits a lognormal distribution better,

statistical analysis is much simpler using a normal distribution.

To further verify this assumption in a more quantitative fashion, a Shapiro-Wilk test of

normality can be done in R to test the null hypothesis that the data is normally distributed [6].

The P-value returned by the Shapiro-Wilk test is 0.06587 for the typing time on the laptop

keyboard, 0.1346 for the typing time on the Nexus keyboard, 0.1364 for the number of typing

errors made on the laptop keyboard, and 0.07519 for the number of typing errors made on the

Nexus keyboard. Thus, under a test with

5% level of significance, we cannot reject the null

hypothesis being that the data is normally distributed, since the P-values for all four sets of data

are greater than 0.05.

Appendix D: Confidence Intervals

The confidence intervals on the means and the standard deviations for the typing times

and error counts are calculated under assumption that the experimental data is normally

distributed (see Appendix C for discussion). The analysis uses a 95% confidence interval, which

results in an

value of 0.05 for one-sided test.

As the population variances are not known, the confidence interval for the means is

calculated using a Student's t-Distribution, as shown in the formula below [4]

̅ −

/2

/√

< < ̅ +

/2

/√

where

̅

is the sample mean,

/2

is the t-value with

- 1 degrees of freedom that gives an area

of

2

⁄

to the right,

is the sample size, and

is the sample variance.

The confidence interval of the standard deviation is calculated using chi-squared distribution, as

shown in the formula below [4]

(−1)

2

2

/2

<

2

<

(−1)

2

2

1−/2

where

is the sample size,

2

is the sample variance, ,

2

/2

and

2

1−/2

are the Chi-squared

values with

−1

degree of freedom that leaves an area of

2

⁄

and ,

1− 2

⁄

respectively, to

the right.