Consider 10 measurements of the diameter of a certain cell type. Assume that the

Question

Consider 10 measurements of the diameter of a certain cell type. Assume that these measurements are random samples drawn from an underlying Gaussian distribution with mean X = 4.38 and standard deviation X = 0.06.

Let Mx be the Random Variable corresponding to the sample mean. a) What kind of a distribution does Mx have?
b) What is the expected value of Mx?
c) What is the standard deviation of Mx?

[Note: this is also called the standard error of the mean (S.E.M)]
d) How does the S.E.M. change if the number of measurements is doubled? In general, if the number of measurements is increased by a factor of n, how does the S.E.M. change? e) What is the 95.4% confidence interval for Mx?

What is the 99.7% confidence interval for Mx?

(Hint: you should not need to use the statistical tables for this part) f) Consider the normalized sample mean Z where:

Z=Mx x x/ n

What kind of a distribution does Z have? Find the 95% confidence interval for Z. (Note: Use the statistical tables provided on the course website)

h) Suppose you find the sample mean for cells in an experiment to be 4.25 inches. Is this sample mean significantly different from the population mean X (at a significance level of 0.05)?
[Show all your steps].

For the remainder of the problem, assume that the standard deviation of the underlying population is unknown but the sample standard deviation Sx is 0.06. Consider the normalized sample mean T where:

T = Mx x Sx / n

i) What kind of a distribution does T have? Find the 95% confidence interval for T. (Note: Use the statistical tables provided on the course website)

j) Suppose you find the sample mean for cells in an experiment to be 4.25 inches. Is this sample mean significantly different from the population mean X (at a significance level of 0.05)?
[Show all your steps].

Explanation / Answer

a) What kind of a distribution does Mx have?

By central limit theorem, Mx is also normally distributed. [ANSWER]

b) What is the expected value of Mx?

By central limit theorem, it has the same mean (expected value),

u(Mx) = 4.38 [ANSWER]

c) What is the standard deviation of Mx?
[Note: this is also called the standard error of the mean (S.E.M)]

It has a reduced standard deviation,

sigma(Mx) = sigma/sqrt(n) = 0.06/sqrt(10) = 0.018973666 [ANSWER]

d) How does the S.E.M. change if the number of measurements is doubled?

As it varies inversely as the square root of n, then it will be reduced by a factor of sqrt(2).

That means it will be divided by 1.4142. [ANSWER]

In general, if the number of measurements is increased by a factor of n, how does the S.E.M. change?

It will be divided by sqrt(n). [ANSWER]

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