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Gannet

Gannet

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A4. Suppose that against a certain opponent the number of points the NUST basketball team scores is normally distributed with unknown mean \( \mu \) and unknown variance \( \sigma^{2} \). Suppose that over a course of 10 games played between the two teams NUST scored the following points 59625974706162666275. (a) Compute the \( 95 \% \) confidence interval for the population mean \( \mu \). \( [3] \) (b) Now suppose that you learn that \( \sigma^{2}=25 \). Compute the \( 95 \% \) confidence interval for the population mean \( \mu \).

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(c) An automatic camera records the number of cars running a red light at an intersection (that is, the cars were going through when the red light was against the car). Analysis of the data shows that on average \( 15 \% \) of light changes record a car running a red light. What is the probability that in 20 light changes there will be exactly three (3) cars running a red light? [5]

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A 5 litre bucket is taken from a swamp. The water contains 75 mosquito larvae. A 200ml flask of water is taken from the bucket for further analysis. What is the probability that the flask contains at least one mosquito larvae?

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\[ [6,14] \] B10. (a) An engineer is interested in the level of carbon dioxide as a waste product from a certain production process. She supposes that the level of carbon dioxide is higher that \( 4.9 \times 10^{-6} \) units. Does a sample of 25 readings with mean \( \bar{x}=5.1 \times 10^{-6} \) units and \( s=2.1 \times 10^{-6} \) present sufficient evidence to sustain the statement? Use the level of significance of \( \alpha=0.05 \).

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6. You are selling a product in an area where \( 30 \% \) of the people live in the city; the rest live in the surburbs. \( 20 \% \) of the city dwellers use your product; and \( 10 \% \) of those from the surburbs use your product. (a) What fraction of the people in the area uses your product? \( [4] \) (b) What proportion of those that use your product are city dwellers. \( [4] \)

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B11. The yield ( \( y \) ) of a chemical process is a random variable whose value is considered to be a linear function of the temperature (x). The following data of corresponding values of \( x \) and \( y \) is found: \begin{tabular}{|c|c|c|c|c|c|} \hline \( \mathrm{x} \) & 0 & 25 & 50 & 75 & 100 \\ \hline \( \mathrm{y} \) & 14 & 38 & 54 & 76 & 95 \\ \hline \end{tabular} (a) Construct a scatter plot of the data and fit an appropriate simple linear regression model. (b) Can a significant relationship between yield and temperature be established? [6] (c) Give the \( 95 \% \) confidence interval of the expected yield at a temperature of \( x_{0}=80 \) degrees celsius.

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(b) A website contains three identical computer servers. Only one is used to operate the site, and the other two are spares that can be activated in case the primary system fails. The probability of a failure in the primary computer from a request for services is 0.005 . Assuming that each request represents an independent trial, what is the mean number of requests until failure of all three spares?

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ANSWERED

Pritesh Ranjan verified

Numerade educator

(b) Let A and B be two events. Suppose the probability that neither A nor B occurs is ( frac{2}{3} ). What is the probability that one or both occur?

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ANSWERED

Ivan Kochetkov verified

Numerade educator

A5. (a) Let C and D be two events with P(C) = 0.25, P(D) = 0.45, and P(C ? D) = 0.1. What is P(C^c ? D)?

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B8. An experiment was conducted to investigate the effect of potassium hydroxide (PH) in the synthesis of biodiesel. It is suspected that potassium hydroxide is related to fatty acid methyl esters (FAME) which are key elements in biodiesel. Three levels of PH concentration were utilised, and six replicates were run in a random order. Data is shown in the following table. (a) Construct the box plots to compare the factor levels and comment on the plot. [4] (b) Construct the analysis of variance. Are there any differences in PH concentration at \( \alpha=0.05 ? \) (c) Analyse the residuals from the experiment and comment if the model is the best fit. (d) Compute the \( 95 \% \) confidence interval on mean FAME when the PH concentration is 1.2

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