TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW08.4

One-sample confidence intervals

(a) Let $I=({\bar {X}}+q_{\alpha /2}\cdot SEM,{\bar {X}}+q_{(1-\alpha /2)}\cdot SEM)$ be Student’s $(1-\alpha )\cdot 100\%$-confidence interval for the expectation, while $q\alpha$ is the $\alpha$-quantile of the $t(n)$-distribution (for $\alpha \in (0,1)$).

Comment on the following statements:

i. At level $\alpha =10\%$ the confidence interval I is smaller than at level $\alpha =$5%

ii. The sample size is $n-1$

iii. If the sample size is doubled, then the width of the confidence interval is halved

iv. The probability to find the expectation left of I is at the most $\alpha$

v. If I overlaps the expectation, then the null hypothesis is not significant.

(b) Messages are regularly sent from a sender to a receiver. For randomly chosen messages the transfer times were measured and stored in the file waitingtimes.Rdata.

i. Represent the data in a histogram. Is the distribution approximately bell-shaped?

ii. Construct an approximate 99%-confidence interval for the expectation and plot it into the graphic.

iii. An engineer claims that the mean transfer time is 1.5 seconds. Are the measurements compatible with this statement?

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

a)

i. At $\alpha =10\%$ the confidence interval $I$ is smaller than at level $\alpha =5\%$.

If we decrease $\alpha$ the quantile $q_{\alpha /2}$ decreases and $q_{1-(\alpha /2)}$ increases, thus widening the interval, so at $10\%$ it is smaller than at $5\%$

ii. The sample size is $n-1$

If we consider a $t(n)$-distribution then $n$ is the degrees of freedom and $n+1$ would be the sample size.

iii. If the sample size is doubled, then the width of the confidence interval is halved

Not necessarily. In most cases the significance only affects the interval size in some proportion but not directly

iv. The probability to find the expectation left of $I$ is at the most $\alpha$

Under $H_{0}:\mu =\mu _{0}$ the probability that C.I. overlaps $\mu _{0}$ is $1-\alpha$. Therefore the probability that $\mu _{0}$ is not within C.I. is $\alpha$

v. If $I$ overlaps the expectation, then the null hypothesis is not significant.

This makes no sense. We don't make any significance statement on the hypothesis. Though we accept $H_{0}$ if $\mu _{0}$ lies within the C.I.

b)

load('./waitingtimes.Rdata')

# i. Represent the data in a histogram. Is the distribution approximately bell-shaped?
hist(wz, breaks = seq(min(wz) - 1, max(wz) + 1, 0.5))

# not bell shaped

# ii.

alpha <- 0.01

length(wz)
m <- mean(wz)
s <- sd(wz)
sem <- s / sqrt(length(wz))
q <- qnorm(1 - (alpha / 2))

lower <- m - q * sem
upper <- m + q * sem

abline(v=m, col='red')
arrows(lower, 15, upper, 15, code=3, length=0.1, angle=90)
text(1.5, 16, "99% - C.I.")
text(m + 0.1, 20, expression(bar(x)), col='red')

# iii.

abline(v=1.5, col='blue')
text(1.6, 20, expression(mu[0]), col='blue')

# it is outside the C.I. so we reject it

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• Waitingtimes.Rdata.zip (details)