Question

Q4.(15) One random sample of the height of one type of tree was recorded.
7, 9 ,10, 7 ,12 ,10, 8, 13 ,15, 9
1.(5) Apply the backward empirical rule to check normality of the data, and conclude if any evidence of non-normality.
2.(5) Assume the height of the tree is normal, calculate the 99% two-sided confidence interval for the true population average height.
3.(5) Assume that we know the population variance σ^2=9,
and we request the bound on the error of estimation of 99% confidence interval to be 2, find the minimum sample size n.

Answer 1

Answer:-

Given That:-

One random sample of the height of one type of tree was recorded.
7, 9 ,10, 7 ,12 ,10, 8, 13 ,15, 9

Given,

Empirical rule:

It is states that for a normal distribution nearly all of the data will fail within three standard deviation of the mean.

The empirical rule can be broken down into three parts.

(i) 68% of data falls within first standard deviation from the mean .

(ii) 95% of data falls within two standard deviation

(iii) 99.7% of data falls within three standard deviation  

Now we discuss given question

First we find mean and standard deviation of sample

n = 10 observations

Standard deviation  

(Using calculator we calculate this values)

1.(5) Apply the backward empirical rule to check normality of the data, and conclude if any evidence of non-normality.

Now we apply empirical rule to check normality

Therefore,

we find

All 10 data points are lies in between

It means all data points are lies in between mean within three standard deviation of the mean.

Therefore,

Given data is normal distributed data.
2.(5) Assume the height of the tree is normal, calculate the 99% two-sided confidence interval for the true population average height.

Now we have to find 99% two sided confidence interval for the true population height.

Formula:

Using z table z_c = 2.5758 ofr 99% confidence inetrval

CI = (10 - 2.138, 10 + 2.138)

CI = (7.862, 12.138)
3.(5) Assume that we know the population variance σ^2=9, and we request the bound on the error of estimation of 99% confidence interval to be 2, find the minimum sample size n.

Given population variance

Population standard deviation =

=  

= 3

Error of estimation = 2

We have to find sample size n

Formula:-

Error of estimation =   (z_c = 2.5758 for 99% CI)

  

Squaring on both sides

n = (3.8637 * 3.8637)

n = 14.93

n = 15 (round off)

Therefore,

Sample size = 15

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