Question

an online article claims that the mean time college students sleep during the week is 6.94 hours with a standard deviation of 1.65 hours. a student is interested in testing if this claim is correct. her two hypotheses are
$h_0: mu = 6.94$
vs
$h_a: mu
eq 6.94$
what type of hypothesis test is this?
o two - tailed
o left - tailed
o right - tailed
she randomly asked 10 students how many hours they slept the night before. their responses were
using technology, what is the sample mean?
$overline{x}=square$
what is the value of the test statistic? $z=square$ (round to 2 decimal places)
which graph represents the corresponding p - value?

Explanation:

Part 1: Type of Hypothesis Test

The alternative hypothesis \( H_a: \mu
eq 6.94 \) indicates that we are testing if the mean is different (either greater or less) from 6.94. This is a two - tailed test because we are interested in deviations in both directions from the hypothesized mean.

Step 1: Sum the data values

The data values are 7, 7, 5, 6, 7, 9, 3, 7, 6, 8. Let's sum them up: \(7 + 7+5 + 6+7 + 9+3 + 7+6 + 8\)

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Step 2: Calculate the sample mean

The sample size \(n = 10\). The sample mean \(\bar{x}=\frac{\text{Sum of data values}}{n}\). So \(\bar{x}=\frac{65}{10}=6.5\)

Step 1: Recall the formula for the z - test statistic

The formula for the z - test statistic in a hypothesis test for the mean (when the population standard deviation \(\sigma\) is known) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)

Step 2: Identify the values

We know that \(\bar{x} = 6.5\), \(\mu=6.94\), \(\sigma = 1.65\), and \(n = 10\)

Step 3: Substitute the values into the formula

First, calculate the denominator \(\frac{\sigma}{\sqrt{n}}=\frac{1.65}{\sqrt{10}}\approx\frac{1.65}{3.1623}\approx0.522\)
Then, calculate the numerator \(\bar{x}-\mu=6.5 - 6.94=- 0.44\)
Now, calculate \(z=\frac{-0.44}{0.522}\approx - 0.84\) (rounded to two decimal places)

Answer:

two - tailed

Part 2: Sample Mean (\(\bar{x}\))