Find two samples when population is normal and Ϭ is known and equal

Example:

A random sample of size 36 from a normal population with variance 24 gave X̅=15 A second sample of size 28 from another normal population with variance 24 give   X̅2=13. test   Ho:μ₁-μ₂=0 against H1: μ₁-μ₂≠0. Let α =0.05 For Two Sided Formulas
Z-test = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)   Z-cal = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)         Z-tab =NORMSINV(1-0.05/2) P-value =2*(1-NORMSDIST(E21))  two samples when population is normal  and Ϭ is known and equal for two sided For One Sided Upper Tail Formulas Z-test = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)   Z-cal = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)   Z-tab =NORMSINV(1-prob) P-value =NORMSDIST(z-cal))       Find two samples when population is normal  and Ϭ is known and equal  for one sided upper tail Lower Tail Formulas Z-test = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)   Z-cal = X̅₁ -X̅₂ /σ*(√1/n₁  +1/n ₂)         Z-tab =NORMSINV(1-prob) P-value =NORMSDIST(z-cal))

Find two samples when population is normal  and Ϭ is known and equal  for one sided lower tail

Find difference of mean of two samples when population is normal and Ϭ is known and unequal on Ms.Excel

Mean of two samples when population is normal and Ϭ is known and unequal:

Let's start admit that this is completely unrealistic, that we will find ourselves in the situation to know the population variance, but not the average population. Therefore,  to learn method of hypothesis testing, we have limited practical application.. As usual, let's start with an example.

Example:

A random sample of size 36 from a normal population with variance 24 gave X̅=15 A second sample of size 28 from another normal population with variance 80 gave X̅2=13.test   Ho:μ₁-μ₂=0 against H1: μ₁-μ₂≠0. Let α =0.05 Solution: For Two sided Test: Z-test = X̅₁  - X̅₂/√(σ₁²/n₁+ σ₂²/n₂)    Z-cal = X̅₁-X̅₂/(sqrt(σ₁ ²/n₁+σ₂ ²/n₂   )          Z-tab = =NORMSINV(1-prob:/2) P-value = =2*(1-NORMSDIST(Z-cal)) Mean of two samples when population is normal and Ϭ is known and unequal for two sided
For One sided Test: Upper Tail: Z-test = X̅₁  - X̅₂/√(σ₁²/n₁+ σ₂²/n₂)    Z-cal = X̅₁-X̅₂/(sqrt(σ₁ ²/n₁+σ₂ ²/n₂   )          Z-tab = =NORMSINV(1-prob:) P-value = NORMSDIST(Z-cal)) Mean of two samples when population is normal and Ϭ is known and unequal for one sided  upper Tail Lower Tail: Z-test = X̅₁  - X̅₂/√(σ₁²/n₁+ σ₂²/n₂)    Z-cal = X̅₁-X̅₂/(sqrt(σ₁ ²/n₁+σ₂ ²/n₂   )          Z-tab = =NORMSINV(1-prob:/2) P-value = =2*(1-NORMSDIST(Z-cal)) Mean of two samples when population is normal and Ϭ is known and unequal for one sided Lower Tail

Find Testing of Mean When population is normal Variance is un known on Ms. Excel

 population is normal Variance is un known:

Which, for purely pedagogical reasons, we have no credible state (a known population variance) is behind us, we turn our attention to the realistic situation of the population means and population differences is unknown.

Now we Explain it with Example:

Example:

Ten individuals are choosen at rendom from a normal population and the hights are found to be inches
63,63,66,67,68,69,70,70,71,and71in the light of these data disscuss the suggestion that mean hight
in population is 66 inches

x

63

63

66

67

68

69

70

70

71

71

mean	67.8
s	3.011091
n	10
d.f	n-1	9

For two sided test

Formulas:

T = X̅- μo/(s/ √n )      

t-cal = X̅- μo/(s/sqrt(n) )      

t-tab =(TINV((Probability,d.f))

p-value=TDIST(ABS(t-cal),d.f,2)

population is normal Variance is un known for two sided

For One sided test:

Upper Tail

Formulas:

T = X̅- μo/(s/ √n )      

t-cal = X̅- μo/(s/sqrt(n) )      

t-tab =(TINV((Probability,d.f))

p-value=TDIST(ABS(t-cal),d.f,1)

1- p value = 1- TDIST(ABS(t-cal),d.f,1)

Decission of C.R =if(t-cal<0,1-p value, p-value)

population is normal Variance is un known for one sided upper tail

Lower Tail

Formulas:

T = X̅- μo/(s/ √n )      

t-cal = X̅- μo/(s/sqrt(n) )      

t-tab =(TINV((Probability,d.f))

p-value=TDIST(ABS(t-cal),d.f,1)

1- p value = 1- TDIST(ABS(t-cal),d.f,1)

Decission of C.R =if(t-cal<0,1-p value, p-value)

population is normal Variance is un known for one sided Lower tail

Find Testing of Mean When population is Normal Variance is Known On Ms. Excel

When population is Normal Variance is Known:

Let's start admit that this is completely unrealistic, that we will find ourselves in the situation to know the population variance, but not the average population. Therefore,  to learn method of hypothesis testing, we have limited practical application. We learn it, just because we will use it to understand hypothesis testing "power" (learning how to calculate the Type II error rate). As usual, let's start with an example.

Example:

A random sample of size 36 is taken from a normal population with a known variance σ² =25
if the mean of the sample is x̅=42.6 ,test the null hypothesis μ=45 against the alternative
hypothesis	μ<45 with α=0.05 (α is the probability of committing Type Ι- error).

For two sided Test:

Formulas:

Z-cal = X̅-μ/(σ/√n) 

Z-tab= = NORMSINV(σ/2)

p-value=  2*(1-NORMSDIST(ABS(Z-cal)))

Now the solution of Example:

When population is Normal Variance is Known for two sided

For One sided Test:

Upper Tail

Z-cal = X̅-μ/(σ/√n) 

Z-tab= = NORMSINV(σ)

p-value= NORMSDIST(ABS(Z-cal))

Now the solution of Example:

When population is Normal Variance is Known for one sided upper tail

Lower Tail

Z-cal = X̅-μ/(σ/√n) 

Z-tab= = NORMSINV(σ)

p-value= NORMSDIST(ABS(Z-cal))

Now the solution of Example:

When population is Normal Variance is Known for one sided lower tail