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2. A simple of 900 students has mean 3.5 cm and standard deviation 2.61 cm is the sample from a large population of mean 3.25 cm and standard deviation 2.61 cm? Check at 99% level of confidence. Also find the range. ​

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Answer:

The sample mean is not from a large population with mean 3.25 cm and standard deviation 2.61 cm

The range is
3.28 < \mu <3.72

Explanation:

From the question we are told that

The sample size is n = 900

The sample mean is
\= x = 3.5 \ cm

The standard deviation is
s = 2.61 \ cm

The population mean is
\mu = 3.25 \ cm

The population standard deviation
\sigma = 2.61 \ cm

Given that confidence level is 99% the significance level is evaluated as


\alpha = (100 - 99)\%


\alpha = 0.01

The critical value of
(\alpha )/(2) is obtained from the normal distribution table , the value is


Z_{(\alpha )/(2) } = 2.58

The null hypothesis
\mu = \= x

The alternative
\mu \\e \= x

Generally the test statistics is mathematically represented


t = (\= x - \mu )/( (s)/( √(n) ) )

=>
t = (3.5 - 3.25)/( (2.61)/( √(900) ) )

=>
t = 2.9

The p-value is from the z-table , the value is


p-value = 2 P(Z > 2.61) = 2 * 0.0018658 = 0.004

So we can see that
p-value < \alpha so we reject the null hypothesis

Hence the sample mean is not from a large population with mean 3.25 cm and standard deviation 2.61 cm

Generally the margin of error is mathematically represented as


E = Z_{(\alpha )/(2) } * (s)/(√(n) )


E = 2.58 * (2.61)/(√(900) )


E = 0.224

The range is evaluated as


\= x - E < \mu < \= x + E

=>
3.5 - 0.224 < \mu < 3.5 + 0.224

=>
3.28 < \mu <3.72

User KimNguyen
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