New Zealand
Level 8 - NCEA Level 3

# Central limit theorem

## Interactive practice questions

The weights of eggs at a particular egg farm are determined to be normally distributed with a mean of $65.7$65.7 grams and a standard deviation of $1.2$1.2 grams.

a

Many egg samples of size $100$100 are taken and measured, and the means of each of the samples, $\overline{X}$X are calculated.

What type of distribution does $\overline{X}$X represent?

Exponential Distribution

A

Discrete Uniform Distribution

B

Normal Distribution

C

Continuous Uniform Distribution

D

Exponential Distribution

A

Discrete Uniform Distribution

B

Normal Distribution

C

Continuous Uniform Distribution

D
b

Calculate the mean of $\overline{X}$X.

c

Calculate the standard deviation of $\overline{X}$X corresponding to a sample size of $100$100. Round your answer to $2$2 decimal places.

Easy
Approx 4 minutes
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Consider a fair $6$6 sided dice, with faces labeled from $1$1 to $6$6. Let $X$X be the outcome when the dice is rolled.

The amount of black coffee dispensed by an automatic coffee machine varies uniformly between $283$283 ml and $311$311 ml.

Let $X$X be the amount of coffee dispensed in a cup.

A discrete random variable $Y$Y has a mean of $50$50 and a standard deviation of $3.6$3.6. Many samples of $52$52 observations are taken and the means $\overline{Y}$Y for each sample calculated.

### Outcomes

#### S8-2

Make inferences from surveys and experiments: A determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem B using methods such as resampling or randomisation to assess

#### 91582

Use statistical methods to make a formal inference