**What are the CRITICAL z-values that correspond to alphas
of 1%, 5% and 10% for the LEFT TAIL and the RIGHT TAIL (6 numbers
required)?**

**Now, split the alphas**

between both ends and give those pairs of values (right and left

tails) for the these

required).

between both ends and give those pairs of values (right and left

tails) for the these

**?**

/2(again, 6 numbers/2

required).

**What are the critical t-test values with 45 df at the**

1%, %5 and 10% levels of significance?

1%, %5 and 10% levels of significance?

Here are graphs showing the Normal (z-values) and the

t-Distributions (t-with more than about 40 degrees of freedom looks

like the Normal). The t or z-value separating the shaded and white

sections can be either our calculated test statistic or the

critical value that we compare the test statistic to (to see if we

are in the “unusual” hence “reject” area).

For now let’s assume the indicated z and t values are the

critical, decision making ones base on the level of significance

(alpha) we have chosen. The values of ? are typically 1%, 5% or 10%

. Graphs “a” and “b” represent ONE-TAILED tests, hence the area in

the shaded region of graph “b” represents the 1%, 5% or 10%

probabilities. We would use this critical t or z value for RIGHT

TAILED tests, where the NULL hypothesis would be something like the

mean is less than or equal to 3.5 (m < 3.5) and the ALTERNATE

hypothesis would be that the mean is greater than 3.5 (m > 3.5).

The NULL MUST have an equality in it.

To find the

**critical value of z**, we go to the

Table and look for the

**z-value**that corresponds to

99% (0.9900), 95% (0.9500) or 90% (0.9000) depending on which alpha

we chose. Then, if our calculated test statistic is a larger number

than the critical value, we would REJECT Ho and accept the

alternate hypothesis. To find the critical t-value, we first need

to calculate the DEGREES OF FREEDOM (but if greater than about 40

we can use the z-table). This all is for a one-tailed test to the

right

If your NULL were that the mean were greater than or equal to

3.5 (ALTERNATE is that the mean is less than 3.5) our critical

value would be in the LEFT TAIL and we would want to determine if

our test statistic is LESS than the critical value (a test

statistic of -0.62 is less than a critical value of -0.61, hence we

would REJECT Ho).

Graphs “c” and “d” are for TWO-TAILED tests in which the NULL

hypothesis is simply that the mean equals 3.5 ( no < or >)

and the ALTERNATE is that the mean is simply NOT EQUAL to 3.5. In

this case we must split the alpha between the two tails. So an

alpha of 5% becomes 2.5% at each tail. So for the right shorter

tail we need an area of 97.5% (0.9750) to the left and for the left

shorter tail we need an area of simply 2.5% (0.2500). For and alpha

of 1% this would be 0.5% (0.0500) at the left tail and 99.5%

(0.9950) to the left to determine the right tail. You can figure

this for an alpha of 10% split between both tails.

Concepts and reasonA critical value is a line on graph that splits the graph into sections. One or two of the sections are called rejection region.The critical value is derived from the significance level, (α)left( alpha right)(α) and the probability distribution of a test statistic. That is, z,t,F,orchi−square.z,t,F,{rm{ or chi – square}}{rm{.}}z,t,F,orchi−square. The

zcritical value is a cut-off point on thezdistribution.Thetcritical value is a cut-off point on thetdistribution. It is almost identical to thezcritical value. The difference is that the shape of thetdistribution is a different from the shape of the normal distribution.FundamentalsThe critical value for lower tail (left tail) area can be found using Excel command, NORMSINV (probability).The critical value for upper tail (right tail) area can be found using Excel command, NORMSINV (1-probability).The critical value of t distribution is obtained from the t tables at a given degrees of freedom with 100(1−α)%100left( {1 – alpha } right)%100(1−α)% level of significance.The critical value of two-sided confidence interval can be found using Excel command T.INV.2T(alpha,degreesoffreedom)T.INV.2Tleft( {{rm{alpha, degrees of freedom}}} right)T.INV.2T(alpha,degreesoffreedom) .The critical value for upper tail is can be found using Excel command T.INV(alpha,degreesoffreedom)T.INVleft( {{rm{alpha, degrees of freedom}}} right)T.INV(alpha,degreesoffreedom) .The critical value for lower tail area can be found using Excel command T.INV(1−alpha,degreesoffreedom)T.INVleft( {{rm{1}} – {rm{alpha, degrees of freedom}}} right)T.INV(1−alpha,degreesoffreedom) .

Step 1

For left tail, the

Zcritical value at 1% significance level is, -2.33 (from standard normal table).The graph is given as follows:For right tail, theZcritical value at 1% significance level is, 2.33 (from standard normal table).The graph is given as follows:For left tail, theZcritical value at 5% significance level is, -1.64 (from standard normal table).The graph is given as follows:For right tail, theZcritical value at 5% significance level is, 1.64 (from standard normal table).The graph is given as follows:For left tail, theZcritical value at 10% significance level is, -1.28 (from standard normal table).The graph is given as follows:For right tail, theZcritical value at 10% significance level is, 1.28 (from standard normal table).The graph is given as follows:Explanation: The critical values can be found in the standard distribution table by corresponding value of the significance level. For 1%, see the probability 0.01, for 5% see the probability 0.05, and for 10% see the probability 0.10. The corresponding row and column is the

Zcritical value.Hint for next step: Use the

tdistribution table to determine thetpercentile of the two-sided confidence interval.Step 2

The critical value table for

tdistribution is given as follows:The significance level is, α=0.01alpha = 0.01α=0.01 and the degrees of freedom are 45.Thet-critical value that is required to construct two-sided confidence interval is, tcrit=tα,df=t0.01,45(fromtcriticalvaluetable)=±2.69begin{array}{c}{t_{crit}} = {t_{alpha ,df}} = {t_{0.01,45}}left( {{rm{from t critical value table}}} right) = pm 2.69end{array}tcrit=tα,df=t0.01,45(fromtcriticalvaluetable)=±2.69 The graph is given as follows:The significance level is, α=0.05alpha = 0.05α=0.05 and the degrees of freedom are 45.Thet-critical value that is required to construct two-sided confidence interval is, tcrit=tα,df=t0.05,45(fromtcriticalvaluetable)=±2.014begin{array}{c}{t_{crit}} = {t_{alpha ,df}} = {t_{0.05,45}}left( {{rm{from t critical value table}}} right) = pm 2.014end{array}tcrit=tα,df=t0.05,45<span class="mop