Question

The following data represent the age in weeks at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. 52 30 44 35 39 26 47 37 56 26 39 28

a. Draw a normal probability plot (use Statcrunch) and boxplot to determine if it is reasonable to conclude the data come from a population that is normally distributed. Copy/paste the graphs into your solution along with an explanation of why or why not.

b. Construct a 90% confidence interval for the mean age at which a baby first crawls.

c. Interpret your confidence interval in part “b”. (I am ____% confident that ______).

d. Sample Size: How large a sample size is needed to estimate the mean age in weeks at which a baby first crawls within 1.5 weeks with 95% confidence? (use formula on pg. 413)

Answer 1

Solution-A:

Rcode:

baby_first_crawls <- c(52, 30, 44, 35, 39, 26, 47, 37, 56 ,26, 39 ,28)
qqnorm(baby_first_crawls)
qqline(baby_first_crawls)
boxplot(baby_first_crawls,main="boxplot",horizontal=TRUE,col="blue"

QQplot:

Boxplot

from qqplot(points are on a straight line) and boxplot (median is didviding the box into two halves)

sample follows normal distribution

we can construct confidence interval for mean

b. Construct a 90% confidence interval for the mean age at which a baby first crawls.

For the given sample

sample mean =

38.25
sample standard deviation=
10.00114
sample size=12
df=n-1=12-1=11

alpha=0.10

alpha/2=0.10/2=0.05

t critical in excel

=T.INV(0.05,11)

=1.79588

90% confidence interval for mean

xbar-t*s/sqrt(n),xbar+t*s/sqrt(n)

38.25-1.79588*10.00114/sqrt(12),38.25+1.79588*10.00114/sqrt(12)

33.06515, 43.43485

90% lower limit mean=33.06515

90% upper limit mean=43.43485

Solution-c:

I am 90% confident that the true mean age at which a baby first crawls lies in between 33.06515 and 43.43485

Solution-d:

n=(Z*s/MOE)^2

z crit fro 95%=1.96

s=sample sd

MOE=1.5

n=(1.96*10.00114/1.5)^2

n= 170.7767

n=171

sample size=n=171