# EXAM 3_SOLUTION

Problems

1. (28 points total, 4 points each) True/False Questions. Circle either True or False.

(a) True or False: The statistic

¯ X−µX S/√n , where ¯ X is the sample mean, µX is the population mean, S is the sample standard deviation, and n is the sample size, is Gaussian-distributed.

(b) True or False: When data X1,...,Xn are Gaussian distributed with variance σ2, and a sample size n has sample variance S2, the statistic (n−1)S2/σ2 has the Chi-squared distribution with n − 1 degrees of freedom.

(c) True or False: If X and Y are independent, then Var [X − Y ] = Var [X] − Var [Y ].

(d) True or False: The sample mean is always equal to the population mean.

(e) True or False: The bias of the sample mean is zero.

(f) True or False: The T distribution is symmetric about zero.

(g) True or False: A “false alarm” is deciding H0 when H1 is true.

2. (24 total points) Assume that random variable X has mean µ and standard deviation σ, but that X is not Gaussian. We can only collect n = 9 samples. Since n < 30, one may not use the CLT to approximate ¯ X as Gaussian. Instead, one may use Chebychev’s theorem to ﬁnd a conﬁdence interval on the population mean of X. If we apply Chebychev’s theorem to the random variable ¯ X (the sample mean), then

P µ ¯ X − kσ ¯ X < ¯X < µ ¯ X + kσ ¯ X ≥ 1 −

1 k2

(1)

Use this expression to develop a 95% conﬁdence interval on µ, as follows.

(a) (8 points) Write expressions for the mean µ ¯ X and standard deviation σ ¯ X of the sample mean ¯ X as functions of µ and σ.

Answer:

(b) (8 points) What k is needed so that the probability that ¯ X is in the interval is ≥ 0.95?

Answer:

(c) (8 points) Rearrange the inequality inside of the probability operator in Equation (1) above so that it is a ≥ 95% conﬁdence interval on µ, with limits that are a function of only ¯ X and σ.

Answer:

3. (30 total points) A communications system manufacturer tests the bit error rate (BER) of receivers they manufacture (under certain test conditions). A sample size of 19 receivers is tested and the sample mean BER is 1.4 × 10−2. The BER is approximately Gaussian distributed.

(a) (12 points) Test the null hypothesis that the population mean is µ = 1×10−2, vs. the alternative hypthothesis that µ is greater than that. They will accept a false alarm rate of 1%, and assume that σ is known to be 0.7 × 10−2. Should H0 be rejected?

(b) (12 points) Now assume that σ is unknown and s = 0.7×10−2. Find a two-sided 95% conﬁdence interval for µ.

(c) (6 points) In part (b), would a 99% conﬁdence interval would be narrower or wider than a 95% conﬁdence interval? Answer “narrower” or “wider”:

4. (18 total points) Let X be a Gaussian random variable. You collect a sample of size n = 21 and ﬁnd the sample mean ¯ x = 12.2 and sample standard deviation s = 4.1.

(a) (12 points) Find a 90% upper one-sided conﬁdence interval on the population standard deviation σ.

(b) (6 points) If n was increased to 31, assuming s remained the same, would the upper limit of the 90% conﬁdence interval increase or decrease? Answer “increase” or “decrease”:

1. (28 points total, 4 points each) True/False Questions. Circle either True or False.

(a) True or False: The statistic

¯ X−µX S/√n , where ¯ X is the sample mean, µX is the population mean, S is the sample standard deviation, and n is the sample size, is Gaussian-distributed.

(b) True or False: When data X1,...,Xn are Gaussian distributed with variance σ2, and a sample size n has sample variance S2, the statistic (n−1)S2/σ2 has the Chi-squared distribution with n − 1 degrees of freedom.

(c) True or False: If X and Y are independent, then Var [X − Y ] = Var [X] − Var [Y ].

(d) True or False: The sample mean is always equal to the population mean.

(e) True or False: The bias of the sample mean is zero.

(f) True or False: The T distribution is symmetric about zero.

(g) True or False: A “false alarm” is deciding H0 when H1 is true.

2. (24 total points) Assume that random variable X has mean µ and standard deviation σ, but that X is not Gaussian. We can only collect n = 9 samples. Since n < 30, one may not use the CLT to approximate ¯ X as Gaussian. Instead, one may use Chebychev’s theorem to ﬁnd a conﬁdence interval on the population mean of X. If we apply Chebychev’s theorem to the random variable ¯ X (the sample mean), then

P µ ¯ X − kσ ¯ X < ¯X < µ ¯ X + kσ ¯ X ≥ 1 −

1 k2

(1)

Use this expression to develop a 95% conﬁdence interval on µ, as follows.

(a) (8 points) Write expressions for the mean µ ¯ X and standard deviation σ ¯ X of the sample mean ¯ X as functions of µ and σ.

Answer:

(b) (8 points) What k is needed so that the probability that ¯ X is in the interval is ≥ 0.95?

Answer:

(c) (8 points) Rearrange the inequality inside of the probability operator in Equation (1) above so that it is a ≥ 95% conﬁdence interval on µ, with limits that are a function of only ¯ X and σ.

Answer:

3. (30 total points) A communications system manufacturer tests the bit error rate (BER) of receivers they manufacture (under certain test conditions). A sample size of 19 receivers is tested and the sample mean BER is 1.4 × 10−2. The BER is approximately Gaussian distributed.

(a) (12 points) Test the null hypothesis that the population mean is µ = 1×10−2, vs. the alternative hypthothesis that µ is greater than that. They will accept a false alarm rate of 1%, and assume that σ is known to be 0.7 × 10−2. Should H0 be rejected?

(b) (12 points) Now assume that σ is unknown and s = 0.7×10−2. Find a two-sided 95% conﬁdence interval for µ.

(c) (6 points) In part (b), would a 99% conﬁdence interval would be narrower or wider than a 95% conﬁdence interval? Answer “narrower” or “wider”:

4. (18 total points) Let X be a Gaussian random variable. You collect a sample of size n = 21 and ﬁnd the sample mean ¯ x = 12.2 and sample standard deviation s = 4.1.

(a) (12 points) Find a 90% upper one-sided conﬁdence interval on the population standard deviation σ.

(b) (6 points) If n was increased to 31, assuming s remained the same, would the upper limit of the 90% conﬁdence interval increase or decrease? Answer “increase” or “decrease”:

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