Explanation

To find the mean, add up the values and divide by the number of values:

0 + 0 + 2 + 2 + 2 + 3 + 3 + 8 + 0 + 10 = 30

There are 10 values.

Mean = 30/10 = 3

Explanation

• Two mutually exclusive events are dependent.
  
• If one event occurs, it completely affects the occurrence of the other, meaning the other event cannot occur, which makes them dependent in this context.

Explanation

Total number of cards: 12

(i) Probability of drawing '8'

• Favorable outcomes: 1 (card number 8)
• Probability: 1/12

(ii) Probability of drawing an even number

• Favorable outcomes: 6 (2, 4, 6, 8, 10, 12)
• Probability: 6/12 = 1/2

(iii) Probability of drawing a perfect square

• Favorable outcomes: 4 (1, 4, 9)
• Probability: 4/12 = 1/3

(iv) Probability of drawing a negative number

• Favorable outcomes: 0 (no negative numbers)
• Probability: 0/12 = 0

(v) Probability of drawing a number less than 13

• Favorable outcomes: 12 (all cards are less than 13)
• Probability: 12/12 = 1

In summary:

• Probability of drawing '8': 1/12
• Probability of drawing an even number: 1/2
• Probability of drawing a perfect square: 1/3
• Probability of drawing a negative number: 0
• Probability of drawing a number less than 13: 1

Explanation

Explanation

A composite index number combines two or more variables into a single index value.

It's often used to measure overall trends, such as in economics or price levels.

Explanation

• The alternate hypothesis is also known as the alternative hypothesis.
• It determines whether a test is one-tailed or two-tailed.

Explanation

• The Rao-Blackwell theorem enables us to obtain minimum variance unbiased estimates through sufficient statistics, which capture all the information in the data about the parameter.
• By conditioning an unbiased estimator on a sufficient statistic, we can improve its precision and obtain the minimum variance unbiased estimator (MVUE).

Explanation

• The variance of a Chi-square distribution with k degrees of freedom is 2k.
• This means that as the degrees of freedom (k) increase, the variance of the Chi-square distribution also increases.
• It indicating greater spread or dispersion in the data.

Explanation

• The Central Limit Theorem (CLT) states that, given certain conditions, the sampling distribution of the sample mean (x̄) will be approximately normal, even if the underlying population distribution is not normal.
• This is true for large sample sizes (typically n > 30).

Explanation

A standard normal distribution has:

Mean = 0

Standard deviation = 1

So, Variance = (Standard deviation)² = 1² = 1