A cereal company claims that its cereal boxes contain an average of $410\text{ g}$ of cereal. A consumer group tests a random sample and constructs a $95\%$ confidence interval for the true mean weight:

$(407.5\text{ g},411.2\text{ g})$

You are given the following null hypothesis:

$H_0:\mu =410\text{ g}$

Does the confidence interval suggest that you should reject $H_0$? Explain your reasoning.

A beverage company claims that the average sugar content in its soft drinks is $30$ grams per bottle. A nutrition researcher collects a random sample and constructs a $95\%$ confidence interval for the mean sugar content: ($29.2$ grams, $29.8$ grams)

You are given the following null hypothesis: $H_0:\mu =30$ grams

Does the confidence interval suggest that you should reject $H_0$? Explain your reasoning.

A battery company claims that its new phone battery lasts at least $12$ hours on average. A testing lab gathers a sample of batteries and constructs a $95\%$ confidence interval for the true average battery life:

($11.3$ hours, $11.9$ hours)

You are given the following null hypothesis:

$H_0:\mu =12$ hours

Does the confidence interval suggest that you should reject $H_0$? Explain your reasoning.

A fitness program is evaluated by measuring the number of push-ups completed by $12$ participants before and after a $6$-week training. The results are:

Construct the $95\%$ confidence interval for the mean difference in push-up counts. What does the interval suggest about the program's effectiveness?

A researcher wants to investigate whether there is a significant difference in the resting heart rates in beats per minute of individuals before and after a $6$-week exercise program. The paired data below represent the heart rates of $8$ participants before and after the program.

Construct a $95$ percent confidence interval for the mean difference in heart rates Before minus After. What does it suggest about the effect of the exercise program?