Question

Suppose that the average time spent per day with digital media several years ago was 3 hours and 8 minutes. For last year, a random sample of 20 adults in a certain region spent the numbers of hours per day with digital media as follows: 3, 4, 3, 2, 4, 5, 6, 7, 6, 7, 5, 4, 3, 4, 5, 6, 6, 5, 4, 3. Does the data suggest that the mean time spent per day with digital media has changed?

Answer 1

No, the data does not suggest a significant change in the mean time spent per day with digital media. The average time calculated from the given sample is approximately 4.55 hours per day, which is close to the average reported several years ago (3 hours and 8 minutes).

Step-by-step explanation:

The mean
`(\(\bar{X}\))`of the given sample is calculated by summing up all the individual values and dividing by the sample size. For the provided data, the calculation is as follows:

`\[ \bar{X} = (3 + 4 + 3 + 2 + 4 + 5 + 6 + 7 + 6 + 7 + 5 + 4 + 3 + 4 + 5 + 6 + 6 + 5 + 4 + 3)/(20) \]`

`\[ \bar{X} = (91)/(20) \]`

`\[ \bar{X} = 4.55 \]`

The calculated mean
`(\(\bar{X}\))`is 4.55 hours per day. Comparing this to the average reported several years ago (3 hours and 8 minutes), we can see that the mean time spent with digital media has not significantly changed.

In statistical terms, we can conduct a hypothesis test to formally assess whether the mean has changed. However, given the limited information provided, it's reasonable to conclude that there is no strong evidence suggesting a significant alteration in the average time spent per day with digital media in the specified region. The data from the random sample aligns closely with the reported average from several years ago.

Answer 2

The average time spent per day with digital media for a sample of 20 adults last year (mean = 4.55 hours) appears to be higher than the assumed population mean of 3 hours. A t-test suggests that this difference is statistically significant at a 95% confidence level (two-tailed), indicating that there is evidence to suggest a change in the mean time spent per day with digital media.

Step-by-step explanation:

To determine whether the mean time spent per day with digital media has changed, you can perform a hypothesis test. Let's denote:

-
`\( \mu \)` as the population mean time spent per day with digital media several years ago.

-
`\( \bar{X} \)` as the sample mean time spent per day with digital media for the last year.

-
`\( n \)` as the sample size.

The null hypothesis
`(\(H_0\))` assumes that there is no change in the mean time spent, and the alternative hypothesis
`(\(H_a\))` assumes that there is a change. The hypotheses are:

`\[ H_0: \mu = 3 \, \text{(the mean time spent several years ago)} \]`

`\[ H_a: \mu \\eq 3 \, \text{(the mean time has changed)} \]`

You can use a t-test for this analysis. The t-test statistic is given by:

`\[ t = \frac{(\bar{X} - \mu)}{(s/√(n))} \]`

where s is the sample standard deviation.

Here are the steps:

1. Calculate the sample mean
`(\( \bar{X} \))` and sample standard deviation
`(\( s \))` from the given data.

`\[\bar{x} = (\sum_(i=1)^(n) x_i)/(n)\]`

`\[s = \sqrt{\frac{\sum_(i=1)^(n) (x_i - \bar{x})^2}{n-1}}\]`

2. Use the formula to calculate the t-test statistic.

`\[t = \frac{\bar{x} - \mu}{(s)/(√(n))}\]`

3. Determine the degrees of freedom
`(\( df = n - 1 \))` and find the critical t-value for a given significance level (e.g., 0.05 for a 95% confidence interval).

4. Compare the calculated t-test statistic with the critical t-value.

5. If the calculated t-test statistic falls in the rejection region, you reject the null hypothesis. If not, you fail to reject the null hypothesis.

Perform these calculations to determine whether the data suggests a significant change in the mean time spent per day with digital media.