Find the critical z values. Assume that the normal distribution applies.

TwoTwo​-tailed ​test; alphaαequals=0.060.06

zequals=

A random sample of 8181 eighth grade​ students’ scores on a national mathematics assessment test has a mean score of 268268 with a standard deviation of 3737. This test result prompts a state school administrator to declare that the mean score for the​ state’s eighth graders on this exam is more than 260260. At alphaαequals=0.120.12​, is there enough evidence to support the​ administrator’s claim? Complete parts​ (a) through​ (e).

​(a) Write the claim mathematically and identify Upper H 0H0 and Upper H Subscript aHa. Choose the correct answer below.

b) Find the standardized test statistic​ z, and its corresponding area.

zequals=

nothing ​(Round to two decimal places as​ needed.)

Areaequals=

nothing ​(Round to three decimal places as​ needed.)

d) Decide whether to reject or fail to reject the null hypothesis.RejectReject Upper H 0H0Fail to rejectFail to reject Upper H 0

At the 12​% significance​ level, there is enough evidence to support the​ administrator’s claim that the mean score for the​ state’s eighth graders on the exam is more than 260.

use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance alphaα​, and sample sizes n 1n1 and n 2n2. Assume that the samples are​ independent, normal, and random. Answer parts​ (a) and​ (b).

Upper H Subscript aHa​: mu 1 less than mu 2μ1<μ2​, alphaαequals=0.050.05​, n 1n1equals=1414​, n 2n2equals=1313

​(a) Find the critical​ value(s) assuming that the population variances are equal.

-1.782

use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance alphaα​, and sample sizes n 1n1 and n 2n2. Assume that the samples are​ independent, normal, and random. Answer parts​ (a) and​ (b).

Upper H Subscript aHa​: mu 1 greater than mu 2μ1>μ2​, alphaαequals=0.0250.025​, n 1n1equals=1313​, n 2n2equals=1111

​(a) Find the critical​ value(s) assuming that the population variances are equal.nothing

​(Type an integer or decimal rounded to three decimal places as needed. Use a comma to 

use the given information to answer parts​ (a) through​ (d).

Upper H 0 : mu 1 equals mu 2H0: μ1=μ2​, alpha equals 0.02α=0.02

Sample​ statistics: x overbar 1 equals 36.2×1=36.2​, s 1 equals 3.7s1=3.7​, n 1 equals 15n1=15

x overbar 2 equals 35.3×2=35.3​, s 2 equals 2.8s2=2.8​, n 2 equals                           8n2=8

Assume sigma Subscript 1 Superscript 2 Baseline equals sigma Subscript 2 Superscript 2σ21=σ22.

negative t 0 equals negative 2.518−t0=−2.518​, t 0 equals 2.518t0=2.518-3

-2

-1

0

1

2

3

t

A normal curve is over a horizontal t-axis labeled from negative 3 to 3 in increments of 1 and is centered on 0. Vertical line segments extend from the horizontal axis to the curve at negative 2.518 and 2.518. The areas under the curve and to the left of negative 2.518 and to the right of 2.518 are shaded.

​(a) Find the test statistic.nothing

rather than the conventional treatment makes a difference in the tensile strength of the bars​ (the ability to resist tearing when pulled​ lengthwise). At alphaαequals=0.050.05​, answer parts​ (a) through​ (e). Assume the population variances are equal and the samples are random. If​ convenient, use technology to solve the problem.

font size decreased by 1 Start 3 By 2 Table 1st Row 1st Column Treatment 2nd Column Tensile strengths left parenthesis newtons per square millimeter right parenthesis 2nd Row 1st Column Experimental 2nd Column Start 1 By 10 Matrix 1st Row 1st Column 365 2nd Column 411 3rd Column 439 4st Column 404 5st Column 380 6st Column 423 7st Column 442 8st Column 9st Column 10st Column EndMatrix 3rd Row 1st Column Conventional 2nd Column Start 1 By 10 Matrix 1st Row 1st Column 393 2nd Column 384 3rd Column 395 4st Column 395 5st Column 380 6st Column 391 7st Column 425 8st Column 433 9st Column 439 10st Column 433 EndMatrix EndTable Treatment Tensile strengths (newtons per square millimeter) Experimental 365 411 439 404 380 423 442 Conventional 393 384 395 395 380 391 425 433 439 433

​(a) Identify the claim and state Upper H 0H0 and Upper H Subscript aHa.

The claim is​ “The new treatment

Which hypothesis is the​ claim?The alternative​ hypothesis, Upper H Subscript aHaThe null​ hypothesis, Upper H 0

The alternative hypothesis is the claim.

b) Find the critical​ value(s) and identify the rejection​ region(s).

Enter the critical​ value(s) below.nothing

use technology to help test the claim about the difference between two population means mu 1μ1 and mu 2μ2 at the given level of significance alphaα using the given sample statistics. Assume that the population is normally​ distributed, and the samples are independent and random. Assume sigma Subscript 1 Superscript 2 Baseline equals sigma Subscript 2 Superscript 2σ21=σ22.

​Claim:

mu 1 less than or equals mu 2μ1≤μ2​; alphaαequals=0.100.10

Find the standardized test statistic.

tequals=

nothing ​(Round to three decimal places as​ needed

Check that you have used the technology correctly. Note that the formula of the standardized test statistic for a​ two-sample t-test for the difference between means when the variances are equal is as shown below.

tequals=StartFraction left parenthesis x overbar 1 minus x overbar 2 right parenthesis minus left parenthesis mu 1 minus mu 2 right parenthesis Over s Subscript x overbar 1 minus x overbar 2 EndFraction

What conditions are necessary in order to use the dependent samples t​-test for the mean of the difference of two​ populations?

Choose the correct answer below.

A.

Each sample must be randomly selected from any population and each member of the first sample must be paired with a member of the second sample.

B.

Each sample must be randomly selected from any population and the two samples must be independent.

C.

Each sample must be randomly selected from a normal population and the two samples must be independent.

D.

Calculate d overbard and s Subscript dsd.

d overbardequals=

nothing

Interpret the decision in the context of the original claim. Choose the correct answer below.

A.

At the​ 1% significance​ level, there isis enough evidence that the​ students’ critical reading scores improved the second time they took the test.

B.

At the​ 1% significance​ level, there is evidence that the​ students’ critical reading scores got worse the second time they took the test.

C.

At the​ 1% significance​ level, there is notis not enough evidence that the​ students’ critical reading scores improved the second time they took the test.

D.

The sample was not large enough to make a conclusion.

The table below shows the gas mileages​ (in miles per​ gallon) of eight cars with and without using a fuel additive. At alphaαequals=​0.10, is there enough evidence to conclude that the fuel additive improved gas​ mileage? Complete parts​ (a) through​ (f).

use the​ t-test to find the standardized test statistic t.

tequals=

nothing

​(Type an integer or a decimal. Round to three decimal places as​ needed

t = -4.536

RejectReject the null hypothesis.Fail to rejectFail to reject the null hypothesis.

the null hypothesis.

f) Interpret the decision in the context of the original claim. Choose the correct answer below.

A.

At the​ 10% significance​ level, there isis enough evidence that the fuel additive improved gas mileage.

B.

At the​ 10% significance​ level, there is notis not enough evidence that the fuel additive improved gas mileage.

C.

At the​ 10% significance​ level, there is enough evidence that the gas mileage was worse when using the fuel additive.

D.

Test the following claim about the difference between two population proportions p 1p1 and p 2p2 for the given level of significance alphaα and the given sample statistics. Is the test​ right-tailed, left-tailed, or​ two-tailed? Assume the sample statistics are from independent random samples.

p 1p1not equals≠p 2p2​, ​Claim:  alphaαequals=0.100.10

x 1x1equals=2121​, n Sample​ statistics:  1n1equals=9696 and x 2x2equals=3636​, n 2n2equals=6363

Is the test​ right-tailed, left-tailed, or​ two-tailed?​Right-tailed​Left-tailed​Two-tailed

two-tailed

Should the null hypothesis be​ rejected?

A.

RejectReject Upper H 0H0. There is sufficientsufficient evidence to support the claim.

B.

RejectReject Upper H 0H0. There is insufficientinsufficient evidence to support the claim.

C.

Fail to rejectFail to reject Upper H 0H0. There is insufficientinsufficient evidence to support the claim.

D.

Fail to rejectFail to reject Upper H 0H0. There is sufficientsufficient evidence to support the claim.

In a study of 18871887 ​adults, 596596 said they has used alternative medicines in the previous year. In a more recent study of 28332833 ​adults, 918918 said they had used alternative medicines in the previous year. At alphaαequals=0.100.10​, can you reject the claim that the proportion of adults using alternative medicines has not changed since the first​ study? Assume the random samples are independent. Complete parts​ (a) through​ (e).

​(a) Identify the claim and state Upper H 0H0 and Upper H Subscript aHa.

The claim is​ “the proportion of adults using alternative medicines has

In a survey of 54455445 male senior​ citizens, 18161816 said they eat the daily recommended number of servings of vegetables. In a survey of 59025902 female senior​ citizens, 19851985 said they eat the daily recommended number of servings of vegetables. At alphaαequals=0.100.10​, can you reject the claim that the proportions of senior citizens who said they eat the daily recommended number of servings of vegetables are the same for the two​ groups? Assume the random samples are independent. Complete parts​ (a) through​ (e).

​(a) Identify the claim and state Upper H 0H0 and Upper H Subscript aHa.

The claim is​ “the proportion of male senior citizens who said they eat the daily recommended number of servings of vegetables is

In a survey of 12061206 adult​ males, 820820 said they use the Internet. In a survey of 19751975 females 14881488 said they use the Internet. At alphaαequals=0.100.10​, can you reject the claim that the proportions of Internet users are the same for the two​ groups? Assume the random samples are independent. Complete parts​ (a) through​ (e).

​(a) Identify the claim and state Upper H 0H0 and Upper H Subscript aHa.

The claim is​ “the proportion of adult male Internet users is

▼

A normal curve is over a horizontal t-axis labeled from negative 3 to 3 in increments of 1 and is centered on 0. Vertical line segments extend from the horizontal axis to the curve at negative 1.708 and 1.708. The areas under the curve and to the left of negative 1.708 and to the right of 1.708 are shaded.

​(a) Find the test statistic.

test statistic = -0.90

Find the standardized test statistic.

tequals=

nothing ​(Type an integer or decimal rounded to three decimal places as​ needed.)

Find the critical​ value(s) for the indicated​ t-test, level of significance alphaα​, and sample size n.

LeftLeft​-tailed ​test, alphaαequals=0.0250.025​, nequals=1717

LOADING… Click the icon to view the​ t-distribution table.

The critical​ value(s) is/are

nothing.

​(Round to the nearest thousandth as needed. Use a comma to separate answers as​ needed.)

each confidence interval indicates that you should reject H0. Explain your reasoning.

An infinite number line, labeled from 0.51 to 0.57, has tick marks in increments of 1. The region to the left of a closed circle at 0.54 is shaded and labeled H@Sub{0}:p≤0.54.

0.51

0.52

0.53

0.54

0.55

0.56

0.57