Question

An important feature of digital cameras is battery​ life, the number of shots that can be taken before the battery needs to be recharged. The accompanying data contains battery life information for 29 subcompact cameras and 16 compact cameras. Complete parts​ (a) through​ (d) below.

Battery life data for the two types of digital camera:

Subcompact Compact

302 394

310 445

289 447

279 260

246 345

197 239

326 332

242 221

276 233

236 256

197 281

223 397

279 507

209 201

261 148

221 129

236

209

208

289

162

276

197

141

232

222

198

168

149

a. Is there evidence of a difference in the variability of the battery life between the two types of digital​ cameras? (Use

alphaαequals=0.05​.)

What are the correct null and alternative​ hypotheses?

What is the test​ statistic?

​(Round to two decimal places as​ needed.)

What is the critical​ value? Select the correct choice below and fill in the answer box within your choice.

​(Round to two decimal places as​ needed.)

A.

Upper F Subscript alphaFαequals=...

B.

Upper F Subscript alpha divided by 2Fα/2equals=...

What is the correct​ conclusion?

A.

Reject

Upper H 0H0.

There is insufficient evidence of a difference in the variability of the battery life between the two types of digital cameras.

B.

Do not reject

Upper H 0H0.

There is insufficient evidence of a difference in the variability of the battery life between the two types of digital cameras.

C.

Reject

Upper H 0H0.

There is sufficient evidence of a difference in the variability of the battery life between the two types of digital cameras.

D.

Do not reject

Upper H 0H0.

There is sufficient evidence of a difference in the variability of the battery life between the two types of digital cameras.

b. Determine the​ p-value in​ (a) and interpret its meaning.

The​ p-value in part​ (a) is.....

​(Round to three decimal places as​ needed.)

What does the​ p-value mean?

A.

The probability of obtaining a sample that yields a test statistic equal to or more extreme than the one in​ (a) is equal to the​ p-value if there is a difference in the two population variances.

B.

The probability of obtaining a sample that yields a test statistic equal to or more extreme than the one in​ (a) is equal to the​ p-value if there is no difference in the two population variances.

C.

The probability of obtaining a sample that yields a test statistic equal to or less extreme than the one in​ (a) is equal to the​ p-value if there is no difference in the two population variances.

D.

The probability of obtaining a sample that yields a test statistic equal to or less extreme than the one in​ (a) is equal to the​ p-value if there is a difference in the two population variances.

c. What assumption about the population distribution of the two types of cameras is necessary in​ (a)?

A.

The populations have equal means.

B.

The populations are the same size.

C.

The populations have different means.

D.

The populations are normally distributed.

Is this assumption​ satisfied?

▼

Yes,

No,

because

▼

the subcompact sample is

the compact sample is

the two samples are

the compact sample mean is

▼

left-skewed.

right-skewed.

smaller than the subcompact sample.

roughly symmetric.

skewed in opposite directions.

smaller than the subcompact sample mean.

equal to the subcompact sample mean.

equal to the subcompact sample.

larger than the subcompact sample mean.

larger than the subcompact sample.

d. Based on the results of​ (a), which t test should be used to compare the mean battery life of the two types of​ cameras?

A.

The​ pooled-variance t test should be​ used, because the two populations have equal variances.

B.

The​ separate-variance t test should be​ used, because the two populations do not have equal variances.

C.

The​ separate-variance t test should be​ used, because the two populations have equal variances.

D.

The​ pooled-variance t test should be​ used, because the two populations do not have equal variances.

Answer 1

a.

Given that,
sample 1
s1^2=12479.63, n1 =16
sample 2
s2^2 =2379.169, n2 =29
null, Ho: sigma^2 = sigma^2
alternate, H1: sigma^2 != sigma^2
level of significance, alpha = 0.05
from standard normal table, two tailed f alpha/2 =2.344
since our test is two-tailed
reject Ho, if F o < -2.344 OR if F o > 2.344
we use test statistic fo = s1^1/ s2^2 =12479.63/2379.169 = 5.245
| fo | =5.245
critical value
the value of |f alpha| at los 0.05 with d.f f(n1-1,n2-1)=f(15,28) is 2.344
we got |fo| =5.245 & | f alpha | =2.344
make decision
hence value of | fo | > | f alpha| and here we reject Ho
ANSWERS
---------------
null, Ho: sigma^2 = sigma^2
alternate, H1: sigma^2 != sigma^2
test statistic: 5.245 =5.24
critical value: -2.34 , 2.34
A.
the value of |f alpha| at los 0.05 with d.f f(n1-1,n2-1)=f(15,28) is 2.34
decision: reject Ho
option:A
we have enough evidence to support the claim that there evidence of a difference in the variability of the battery life between the two types of digital​ cameras

b.
Given that,
population variance (sigma^2) =12479.63
sample size (n) = 16
sample variance (s^2)=2379.169
null, Ho: sigma^2 =12479.63
alternate, H1 : sigma^2 !=12479.63
level of significance, alpha = 0.05
from standard normal table, two tailed chisqr^2 alpha/2 =24.996
since our test is two-tailed
reject Ho, if chisqr^2 o < - OR if chisqr^2 o > 24.996
we use test statistic chisquare chisqr^2 =(n-1)*s^2/o^2
chisqr^2 cal=(16 - 1 ) * 2379.169 / 12479.63 = 15*2379.169/12479.63 = 2.86
| chisqr^2 cal | =2.86
critical value
the value of |chisqr^2 alpha| at los 0.05 with d.f (n-1)=15 is 24.996
we got | chisqr^2| =2.86 & | chisqr^2 alpha | =24.996
make decision
hence value of | chisqr^2 cal | < | chisqr^2 alpha | and here we do not reject Ho
chisqr^2 p_value =0.9997
ANSWERS
---------------
null, Ho: sigma^2 =12479.63
alternate, H1 : sigma^2 !=12479.63
test statistic: 2.86
critical value: -24.996 , 24.996
p-value:0.9997
decision: do not reject Ho
option:A
The probability of obtaining a sample that yields a test statistic equal to or more extreme than
the one in​ (a) is equal to the​ p-value if there is a difference in the two population variances

c.
assumption about the population distribution of the two types of cameras is necessary in​ (a)
option:C
The populations have different means.
assumption are satisfied because two samples are larger than the subcompact sample mean.

d.
Based on the results of​ (a),
t test should be used to compare the mean battery life of the two types of​ cameras
option:D
The​ pooled-variance t test should be​ used, because the two populations do not have equal variances