3. The viscosity of a chemical product is read every 2 minutes. Some data from this...

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3. The viscosity of a chemical product is read every 2 minutes. Some data from this process can be found in the viscosity.csv file, which has only one variable, viscosity, measured in micropascal per second units (μPa.s). (a) Use t.test() to construct a 95% confidence interval for the mean. [1 m [3 ma (b) We will now construct a 95% bootstrap confidence interval for the mean. To do so, we first need to create an empty numeric vector, then use a for loop to do the following 1000 times: i. Use dplyr::slice_sample () to sample the original data with replacement. Each new sample should have the same sample size (number of observations) as the original data. ii. Calculate the sample mean of the resampled data. iii. Save the sample mean as the ith element of the numeric vector created prior to the for loop. The result of this for loop creates what is known as a bootstrap distribution of sample means, which is centred on the observed sample mean. (c) Calculate the bootstrap distribution's 2.5% and 97.5% percentiles (quantiles). These [2 ma correspond to the 95% bootstrap confidence interval for the mean. Ensure that these values are printed out in your Quarto document. (d) Visualise the bootstrap distribution and superimpose the percentiles calculated in part [2 ma (c) as reference lines using the ggplot2 package. Ensure that your plot at least has appropriate axis labels. Hint: Consider using the ggplot2::geom_hline () or ggplot2:: geom_vline () commands (functions) to draw the reference lines. (e) Compare the standard 95% confidence interval for the mean against the 95% bootstrap [3 ma confidence interval for the mean. Hence, describe which statistical concept the bootstrap distribution approximates by resampling the data. 3. The viscosity of a chemical product is read every 2 minutes. Some data from this process can be found in the viscosity.csv file, which has only one variable, viscosity, measured in micropascal per second units (μPa.s). (a) Use t.test() to construct a 95% confidence interval for the mean. [1 m [3 ma (b) We will now construct a 95% bootstrap confidence interval for the mean. To do so, we first need to create an empty numeric vector, then use a for loop to do the following 1000 times: i. Use dplyr::slice_sample () to sample the original data with replacement. Each new sample should have the same sample size (number of observations) as the original data. ii. Calculate the sample mean of the resampled data. iii. Save the sample mean as the ith element of the numeric vector created prior to the for loop. The result of this for loop creates what is known as a bootstrap distribution of sample means, which is centred on the observed sample mean. (c) Calculate the bootstrap distribution's 2.5% and 97.5% percentiles (quantiles). These [2 ma correspond to the 95% bootstrap confidence interval for the mean. Ensure that these values are printed out in your Quarto document. (d) Visualise the bootstrap distribution and superimpose the percentiles calculated in part [2 ma (c) as reference lines using the ggplot2 package. Ensure that your plot at least has appropriate axis labels. Hint: Consider using the ggplot2::geom_hline () or ggplot2:: geom_vline () commands (functions) to draw the reference lines. (e) Compare the standard 95% confidence interval for the mean against the 95% bootstrap [3 ma confidence interval for the mean. Hence, describe which statistical concept the bootstrap distribution approximates by resampling the data.

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