Question

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of $22$LTCC Intermediate Algebra students generated a mean of $7.24$hours with a standard deviation of $1.93$hours. At a level of significance of $5\%$, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?

The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours

$a.$is more than seven hours.

$b.$is at most seven hours.

$c.$is at least seven hours.

$d.$is less than seven hours.

Short answer

Type II error is the mean number of hours of sleep students get per night is a minimum of seven hours when, in fact, it really is a smaller amount than seven hours. Correct option is d.

Step-by-step solution

Given Information

LTCC Intermediate Algebra students get but seven hours of sleep per night, on average.

The results for a test are as follows:

$H_0:\mu \ge 7\mathrm{h},H_a:p<7\mathrm{h}$

at level of significance of $\alpha =5\%.$The $p-value$for the test is

$p-value=P\{\overline{x}>7.24\}=P\{\overline{x}>7.24\}=0.28$

When the p-value is larger than the established alpha value we don't reject the null hypothesis.

Now, we see that

$\text{p-value}=0.28>0.05=\alpha$

Explanation Part a

Therefore, we don't reject $H_0:\ge 7h$

We don't have enough evidence to cast serious doubt on the truthfulness of $H_0$.

At tier significance of $5\%$, LTCC Intermediate Algebra students get a minimum of seven hours of sleep per night, on average.

Rejecting the null hypothesis $H_0$when it's true is defined as a type I error If the $H_0$is: "LTCC Intermediate Algebra students get a minimum of seven hours of sleep per night, on average," then a Type I error is

We conclude that the mean number of hours of sleep students get per night is smaller amount than seven hours when, in fact, it really is a minimum of seven hours.

Explanation Part b

The mean number of hours of sleep students get per night is a minimum of seven hours when, in fact, it really is a smaller amount than seven hours therefore, we can not say that at most seven hours of sleep is correct option

Explanation Part c

The mean number of hours of sleep students get per night is a minimum of seven hours when, in fact, it really is a smaller amount than seven hours therefore, we can not say that at least seven hours of sleep is correct option.

Explanation Part d

Failing to reject the null hypothesis when it's false is defined as a type II error.

Therefore, Type II error is

We conclude that the mean number of hours of sleep students get per night is a minimum of seven hours when, in fact, it really is a smaller amount than seven hours.