# Null and alternative hypothesis (1 Viewer)

#### Hivaclibtibcharkwa

##### 𝗕𝗶𝗼𝗹𝗼𝗴𝘆 𝗧𝘂𝘁𝗼𝗿
Can someone help me with these question, like with question b would the null hypothesis be that it’s less than 70 or would the alternative hypothesis say that?

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#### cossine

##### Active Member
Can someone help me with these question, like with question b would the null hypothesis be that it’s less than 70 or would the alternative hypothesis say that?
The null hypothesis is always an equality. The reason for this is computation.

Because the definition p-value is

p-value = P(getting test statistic that is more contradictory or equally contradictory to the null hypothesis | H0 )

It may help to read Probability and Statistics for Engineering Sciences by Jay L. Devore

It will explain things in more detail

#### Hivaclibtibcharkwa

##### 𝗕𝗶𝗼𝗹𝗼𝗴𝘆 𝗧𝘂𝘁𝗼𝗿
The null hypothesis is always an equality. The reason for this is computation.

Because the definition p-value is

p-value = P(getting test statistic that is more contradictory or equally contradictory to the null hypothesis | H0 )

It may help to read Probability and Statistics for Engineering Sciences by Jay L. Devore

It will explain things in more detail
Ah sorry just realised I didn’t put in the imahe

#### cossine

##### Active Member
Ah sorry just realised I didn’t put in the imahe
So give the question a go first and I will check the working out.

In this case

H0: mean_weight = 70
Ha: mean_weight <= 70.

#### jimmysmith560

##### Le Phénix Trilingue
Moderator
In general, we reject the null hypothesis if the p-value is less than the significance level α.

If the alternative hypothesis $\bg_white H_a$ contains the not‐equal‐to symbol (≠), the hypothesis test is a two‐tailed test. Otherwise, all other tests are one-tailed.

If needed, below are solutions to each part, based on the above information. As cossine mentioned, definitely try to attempt the question(s) first to see if you can apply your understanding in your working:

Part (a):

Suppose $\bg_white \mu$ is the average age of Sydney residents; then $\bg_white H_0$ is $\bg_white \mu=35$, and $\bg_white H_a$ is $\bg_white \mu>35$. Note that this is a one-tailed hypothesis test. Here, the p-value 0.0170 is less than the significance level $\bg_white \alpha=0.05$. So we reject the null hypothesis.

Conclusion: this sample information does indicate that the mean age in Sydney has increased from 35 years.

Part (b):

Let $\bg_white \mu$ be the average weight of this population; then $\bg_white H_0$ is $\bg_white \mu=70$, and $\bg_white H_a$ is $\bg_white \mu<70$. This is a one-tailed test. The standard deviation of weights in this population is $\bg_white \sqrt{49}=7$. Here, the p-value 0.001 is less than the significance level $\bg_white \alpha=0.01$. So we reject the null hypothesis.

Conclusion: the sample data do provide enough evidence for us to conclude that the mean weight for the population is less than 70 kg.

Part (c):

Let $\bg_white \mu$ be the average IQ score of this population; then $\bg_white H_0$ is $\bg_white \mu=100$, and $\bg_white H_a$ is $\bg_white \mu \ne 100$. This is a two-tailed test. Here, the p-value 0.095 is greater than the significance level $\bg_white \alpha=0.05$. So we do not reject the null hypothesis.

Conclusion: on the basis of these data, we cannot conclude that the mean IQ score for this population is not 100.

I hope this helps!

#### Hivaclibtibcharkwa

##### 𝗕𝗶𝗼𝗹𝗼𝗴𝘆 𝗧𝘂𝘁𝗼𝗿
The null hypothesis is always an equality. The reason for this is computation.

Because the definition p-value is

p-value = P(getting test statistic that is more contradictory or equally contradictory to the null hypothesis | H0 )

It may help to read Probability and Statistics for Engineering Sciences by Jay L. Devore

It will explain things in more detail
Can it also be the greater than or equal to symbol?
3:37 of this video

#### cossine

##### Active Member
Can it also be the greater than or equal to symbol?
3:37 of this video
So if you have

H0: mean_weight >= 70
Ha: mean_weight <70

Reject the null hypothesis would be equivalent to rejecting H0: mean_weight=70

For computing the z-value you would have mean_weight=70.

#### 5uckerberg

##### Well-Known Member
Can someone help me with these question, like with question b would the null hypothesis be that it’s less than 70 or would the alternative hypothesis say that?View attachment 35496
IIRC to determine null and alternative hypothesis the alternative hypothesis is when this statement is true null hypothesis is when it is not.

#### 5uckerberg

##### Well-Known Member
So give the question a go first and I will check the working out.

In this case

H0: mean_weight = 70
Ha: mean_weight <= 70.
I believe for accuracy you should put a tilde above $\bg_white H_{0}$ because $\bg_white H_{0}:\mu=70$ is the boundary condition for the question. Like this $\bg_white \widetilde{H_{0}}:\mu=70$

#### 5uckerberg

##### Well-Known Member
In general, we reject the null hypothesis if the p-value is less than the significance level α.

If the alternative hypothesis $\bg_white H_a$ contains the not‐equal‐to symbol (≠), the hypothesis test is a two‐tailed test. Otherwise, all other tests are one-tailed.

If needed, below are solutions to each part, based on the above information. As cossine mentioned, definitely try to attempt the question(s) first to see if you can apply your understanding in your working:

Part (a):

Suppose $\bg_white \mu$ is the average age of Sydney residents; then $\bg_white H_0$ is $\bg_white \mu=35$, and $\bg_white H_a$ is $\bg_white \mu>35$. Note that this is a one-tailed hypothesis test. Here, the p-value 0.0170 is less than the significance level $\bg_white \alpha=0.05$. So we reject the null hypothesis.

Conclusion: this sample information does indicate that the mean age in Sydney has increased from 35 years.

Part (b):

Let $\bg_white \mu$ be the average weight of this population; then $\bg_white H_0$ is $\bg_white \mu=70$, and $\bg_white H_a$ is $\bg_white \mu<70$. This is a one-tailed test. The standard deviation of weights in this population is $\bg_white \sqrt{49}=7$. Here, the p-value 0.001 is less than the significance level $\bg_white \alpha=0.01$. So we reject the null hypothesis.

Conclusion: the sample data do provide enough evidence for us to conclude that the mean weight for the population is less than 70 kg.

Part (c):

Let $\bg_white \mu$ be the average IQ score of this population; then $\bg_white H_0$ is $\bg_white \mu=100$, and $\bg_white H_a$ is $\bg_white \mu \ne 100$. This is a two-tailed test. Here, the p-value 0.095 is greater than the significance level $\bg_white \alpha=0.05$. So we do not reject the null hypothesis.

Conclusion: on the basis of these data, we cannot conclude that the mean IQ score for this population is not 100.

I hope this helps!
Now one-tailed test. That is where you need to know what the alternative hypothesis is pointing at and then you will have the P-value pointing in the same direction towards the test statistic which would be found using a modified z-score technique.

#### cossine

##### Active Member
I believe for accuracy you should put a tilde above $\bg_white H_{0}$ because $\bg_white H_{0}:\mu=70$ is the boundary condition for the question. Like this $\bg_white \widetilde{H_{0}}:\mu=70$
I have never seen such notation before. Do you have a reference?