A score of 243 is a "basic" reading level.
Scores for a random sample of 3000 students had a xbar = 250 with a standard error 1.0
The 99% confidence interval for the mean score all students in the city is 247.4 to 252.6.
Ques: Is there good evidence that the mean for all students in the city is less than the basic level?
Ans: 243 is not in the 99% confidence level. It is below.
The answer in the back of the book says:"we would fail to reject Ho: u=243 against the one-sided alternative hypothesis Ha: u (less than) 243 at the 1% significance level."
I do not understand the book answer "we would fail to reject..." What does fail to reject mean?
Does this mean there IS good evidence that the mean is less than the basic level? or Not?
Your point is well taken. Sometimes academics tend to write in meaningless, ill-defined phrases. Here, it is clear the hypothesis is rejected.
So there is NOT good evidence, correct?
THERE IS NOT GOOD EVIDENCE
When performing a hypothesis test, we start by assuming a null hypothesis (Ho) and an alternative hypothesis (Ha). In this case, the null hypothesis is that the mean score for all students in the city is equal to the basic level of 243, and the alternative hypothesis is that the mean is less than 243.
To determine whether there is evidence to support the alternative hypothesis, we conduct a hypothesis test using the given information. The book answer states "we would fail to reject Ho: u=243 against the one-sided alternative hypothesis Ha: u (less than) 243 at the 1% significance level."
"Failing to reject" the null hypothesis means that we do not have sufficient evidence to support the alternative hypothesis. In other words, we do not have enough evidence to conclude that the true mean score is actually less than the basic level.
In this case, if the mean score falls within the 99% confidence interval (247.4 to 252.6), it means that the observed sample mean of 250 is not significantly different from the basic level of 243. Therefore, we cannot confidently claim that the mean score is less than the basic level.
To summarize, based on the information provided, there is not enough evidence to conclude that the mean for all students in the city is less than the basic level.