Let's analyze each pair of hypotheses to determine whether they are correctly set up as the null and alternative hypotheses for a hypothesis test.
1. Hypotheses:
- [tex]$H_0: \mu \neq 344$[/tex]
- [tex]$H_1: \mu = 344$[/tex]
The null hypothesis ([tex]$H_0$[/tex]) should always include an equality sign (either "=", "≤", or "≥"). In this case, [tex]$H_0$[/tex] contains a "≠" sign, which is incorrect because the null hypothesis should represent a statement of no effect or no difference. Hence, this setup is Invalid.
Answer: Invalid
2. Hypotheses:
- [tex]$H_0: \bar{X} = 165$[/tex]
- [tex]$H_1: \bar{X} \neq 165$[/tex]
The null hypothesis should concern the population parameter, not a sample statistic. Here, [tex]$H_0$[/tex] states that the sample mean [tex]$\bar{X}$[/tex] equals 165 instead of the population mean [tex]$\mu$[/tex]. This makes the hypothesis setup incorrect. Therefore, this setup is Invalid.
Answer: Invalid
3. Hypotheses:
- [tex]$H_0: \sigma = 79$[/tex]
- [tex]$H_1: \sigma \neq 79$[/tex]
In this case, the null hypothesis states that the population standard deviation [tex]$\sigma$[/tex] equals 79, and the alternative hypothesis states that [tex]$\sigma$[/tex] is not equal to 79. Both statements involve the population parameter, and the null hypothesis includes an equality sign, as required. Hence, this setup is Valid.
Answer: Valid
4. Hypotheses:
- [tex]$H_0: \mu = 39$[/tex]
- [tex]$H_1: \mu \neq 39$[/tex]
Here, the null hypothesis states that the population mean [tex]$\mu$[/tex] equals 39, and the alternative hypothesis states that [tex]$\mu$[/tex] is not equal to 39. Both statements involve the population parameter, and the null hypothesis includes an equality sign, as required. Therefore, this setup is Valid.
Answer: Valid
In conclusion, the answers are as follows:
1. Invalid
2. Invalid
3. Valid
4. Valid
