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Basic Statistics

If A and B are two events such that Pr(A) = 0.7 and P( A n B)=0.3, we can then conclude that:

A and B are two events such that Pr(A) = 0.5, Pr(B) = 0.2 and P(A n B)=0.1. Therefore, we can then conclude that:

Of the A-level students at a certain school, 60% of those taking Mathematics also take Economics, while 30% of those not taking Mathematics take Economics. Overall, 35% of the A-level students at this school take Mathematics. A student is selected randomly from those taking Economics. The probability that this student is also studying Mathematics is:

The random variable W=2X –Y+3Z+6 where X, Y and Z are three independent random variables. The expectations and variances of these are given by: E(X) = 3, var(X) = 2, E(Y) = -2, var(Y) = 3, E(Z) = 1, var(Z) = 1.5. The expected value of W is equal to:

The random variable W=2X –Y+3Z+6 where X, Y and Z are three independent random variables. The expectations and variances of these are given by: E(X) = 3, var(X) = 2, E(Y) = -2, var(Y) = 3, E(Z) = 1, var(Z) = 1.5. The variance of W is equal to:

Suppose that, after taking logarithms to base e, annual household income (measured in £s) is normally distributed with mean 8.6 and variance 1.44. The probability that a randomly chosen household has an income greater than £75,000 per annum is:

A Social Scientist is interested in the wages of married couples. She knows that the hourly wages of wives are normally distributed with a mean of 8.75 Euros and a variance of 2.31. The hourly wages of husbands are normally distributed with a mean of 9.53 Euros and a variance of 3.82. If there is a positive covariance of 0.82 between the wages of husbands and wives, for what proportion of married couples will the wife earn at least 2 Euros per hour less than her husband (to 2 decimal places)?

A random variable X has mean ~\mu~ and variance ~\sigma^2~. A random sample of 2 observations is to be taken from the probability distribution of X, yielding the sample random variables ~X_1~ and ~X_2~. Four estimators of μ are: ~G=(X_1+X_2)/2~,~H=X_1,~J=0.3X_1+0.7X_2,~K=0.62X_1+0.35X_2~. Which of these estimators is a biased estimator of ~\mu~?

A random variable X has mean ~\mu~ and variance ~\sigma^2~. A random sample of 2 observations is to be taken from the probability distribution of X, yielding the sample random variables ~X_1~ and ~X_2~. Four estimators of ~\mu~; are: ~G=(X_1+X_2)/2~,~H=X_1,~J=0.3X_1+0.7X_2,~K=0.62X_1+0.35X_2~. Which of the unbiased estimators is the most efficient?

You draw a random sample of size 2 from the probability distribution of the random variable X, which is given in the table
 
\begin{tabular}{llll}
X~&~-1~&~0~&~1~\\
Pr(.)&~1/6~&~2/6~&~3/6~\\
\end{tabular}
What are the probabilities of drawing the samples (-1,0), (1,-1), and (1,0)?

You draw a random sample of size 2 from the probability distribution of the random variable X, which is given in the table
 
\begin{tabular}{llll}
X~&~-1~&~0~&~1~\\
Pr()~&~1/6~&~2/6~&~3/6~\\
\end{tabular}
What are the values the sample mean, ~\bar{X}, can take:

Using the information given above. Calculate the expected value of the sample mean, ~E(\bar{X}).

Using the information given above. Calculate the variance of the sample mean, ~V(\bar{X}).

The random variable X with probability distribution N(4, 0.4). What is the value of x which satisfies Pr(X>x)=0.01?

A recent survey of 853 teenagers in San Francisco reported that 47.01% of them felt that Twitter is cooler now than it was one year previously. This information is to be used to test the hypothesis that at least 50% of all teenagers in San Francisco have this view, against a one-tailed alternative hypothesis. What is the value of the test statistic for the hypothesis test?

A research laboratory is checking the carbon emissions of a certain type of engine and these are known to follow a normal distribution. A random sample of 10 engines is tested, yielding the following summary statistics:
 
\begin{tabular}{llllll}
$Variable$~&~$N$~&~$mean$~&~$variance$~&~$sd$~&~$se(mean)$~\\
$Emit$~&~10~&~17.17~&~8.889    &~2.981442~&~0.9428149~\\
\end{tabular}
The legal maximum level of the carbon emissions is 20 (parts per million). Test the hypothesis that the population mean level of carbon emissions for this type of engine is 20, against the alternative hypothesis that the population mean is less than 20. Use a Type I error probability of 1%. What is the value of the test statistic?

What is the critical value for the test in the question above?

What does the result of the test given 2 questions earlier and the critical value from the previous question tell you about the carbon emissions for this type of engine? Choose just one of the alternatives that best reflects these properties.

At the 1% significance level calculate the power of the test that the population mean level of carbon emissions for this type of engine is 20, against the alternative hypothesis that the population mean is less than 20. Given that the true mean is 17.

The joint distribution of the pair of random variables as given in the following table:
 
\begin{tabular}{lllll}
~&~&    &~$X$~&~\\
~&~&~1&~2~&~3~\\
$Y$&~-1~&~0.3~&~0.2~&~0.1~\\
&~1~&~0.2~&~0.1~&~0.1~\\
\end{tabular}
What is the covariance between X and Y?

A random sample of size five, for two variables X and Y has been collected. The values for the data are:
 
\begin{tabular}{lll}
&~$X$~&~$Y$~\\
1~&~7~&~15~\\
2~&~8~&~13~\\
3~&~10~&~10~\\
4~&~12~&~7~\\
5~&~13~&~5~\\
\end{tabular}
The sample covariance between X and Y is:

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