Mathematics 243
(18.16)
We are assuming that a packet behaves like a random
sample of all seeds (Randomization Condition) and that
the package is less than 10% of the whole population
(10% Condition). This latter is true since the population is all possible seeds produced by this company. If
we use the normal approximation, we must also check
the Success/Failure condition (8% of 160 is 13). We can
compute this probability exactly using the binomial or
approximately using the normal approximation.
> 1-pnorm(.95,.92,sqrt(.92*.08/160))
[1] 0.0809429
> 1-pbinom(.95*160,160,.92)
[1] 0.05275305
(18.18)
Computation using both the binomial and the normal approximation to the binomial.
March 19 Homework Solutions
that there is a 90% probability that this particular interval contains the parameter. Our confidence is in the
procedure, not the particular interval. 90% of all 90%
confidence intervals will contain the true value of the parameter.
(19.24)
(a) Randomization Condition: random sample is stated.
10% condition: There are many more that 17,000 college
freshmen. Success failure condition: 74% of 1644 and
26% of 1644 are both greater than 10.
(b)
> phat=.74
> se = sqrt( .74*.26/1644)
> zcrit = qnorm(.99)
> c( phat- zcrit*se, phat+ zcrit*se)
[1] 0.7148333 0.7651667
(c) We are 98% confident that the national freshmen-tosophomore retention rate is between 71.5% and 76.5%.
(d) In 98% of random samples, the 98% confidence interval captures the true proportion so we can be 98$ confident that this interval does.
(19.28)
(19.22)
(a) The parameter being estimated is the percentage of
legal songs in the digital music libraries of all students.
> phat=49/207
The population is songs and the sample size is 117,079.
> se=sqrt ( phat * (1-phat) /207)
(b) Certainly the 10% Condition and Success failure con> zcrit = qnorm(.95)
dition are satisfied. The songs sampled are not a simple
> c( phat- zcrit*se, phat+ zcrit*se)
random sample but rather a cluster sample of all songs.
[1] 0.1881192 0.2853107
This means that independence is rather badly violated.
(a) A 90% confidence interval for the success rate of this (c)
clinic is (.188, .285).
> phat=.231
(b) We can be 90% confident that the true success rate > se = sqrt ( phat * (1-phat) /117079)
of the clinic is in this interval.
> zcrit=qnorm(.975)
(c) 90% of all confidence intervals constructed by this > c( phat- zcrit*se, phat+ zcrit*se)
method capture the true value of p.
[1] 0.2285858 0.2334142
(d) It is consistent with this interval that the true success
rate is 25% as 25% is in the interval.
(d) We can be 95% confident that between 22.9% and
23.3% of all student songs are legal. While this confiIt is very easy to incorrectly describe what a confidence dence interval was correctly constructed, it is quite small
interval means. Read page 472 to get a sense of what due to the large sample (and subsequent small standard
a confidence interval does not mean! It is important to error). We wouldn’t necessarily trust it to this level of
realize that a probability statement is always about the confidence especially because of the cluster sample rather
future. So don’t say that a 90% confidence interval means than the SRS.
> 1-pbinom(19,732,.04)
[1] 0.973135
> 1-pnorm(19/732,.04, sqrt(.04*.96/732))
[1] 0.9737477