are the following statements, Ho:=10 and Ha:equal to or less than, a valid pair of null and alternative hypothesis?assume that in a hypothesis test with null hypothesis Ho : u = 12.0 at a = 0.10, that a value of 16.0 for the sample mean results in the null hypothesis being rejected. that corresponds to a confidence interval result of

It would help if you proofread your questions before you posted them.

Ho: mean = 10
Ha: mean < 10 ("equal to" is the Ho.)

"Ho : u = 12.0 at a = 0.10"?

To determine if the given statements are a valid pair of null and alternative hypotheses, we need to understand the structure of these hypotheses.

The null hypothesis (Ho) is a statement of no effect or no difference, typically denoted with an equals sign (=) or an inequality. It assumes that there is no significant relationship or difference between parameters.

The alternative hypothesis (Ha), on the other hand, represents the opposite of the null hypothesis. It suggests that there is a significant relationship or difference. It can be one-sided, indicated by "<" or ">", or two-sided, indicated by "≠" (not equal to).

In the given statements:

Ho := 10 (null hypothesis: u = 10.0)
Ha: equal to or less than (alternative hypothesis: u ≤ 10.0)

These are a valid pair of null and alternative hypotheses because Ho assumes that the population mean (u) is equal to 10.0, while Ha suggests that the population mean (u) can be equal to or less than 10.0.

Now, let's address the second part of your question about hypothesis testing and confidence intervals.

When performing a hypothesis test with a null hypothesis Ho: u = 12.0 at a significance level (α) of 0.10, a value of 16.0 for the sample mean results in rejecting the null hypothesis.

A confidence interval is a range of values within which the population parameter is likely to fall. In this case, since the null hypothesis is rejected, it means that the true population mean is unlikely to be 12.0.

Unfortunately, the question you asked seems incomplete, as it cuts off before providing the details about the confidence interval result. Consequently, I cannot provide an explanation for the confidence interval result. If you have any additional information or further questions, please let me know, and I'll be happy to assist you.