The the data show the number of hours per day 12 adults spent infront of screens watching television content and these selected form normal distribution
2 5 4 6 7
4 2 3 2 3
Construct a 95% confidence interval to estimate the average number of hours per day adults spend in watching television.
To construct a 95% confidence interval, we will use the t-distribution since the population standard deviation is unknown.
First, we need to calculate the sample mean and standard deviation:
Sample mean (x̄) = (2+5+4+6+7+4+2+3+2+3)/10 = 3.8
Sample standard deviation (s) = √[(Σ(x - x̄)²)/(n-1)]
= √[((2-3.8)² + (5-3.8)² + (4-3.8)² + (6-3.8)² + (7-3.8)² + (4-3.8)² + (2-3.8)² + (3-3.8)² + (2-3.8)² + (3-3.8)²)/(10-1))]
= 1.623
Next, we need to determine the t-value for a 95% confidence interval with 9 degrees of freedom (n-1=10-1). From the t-distribution table, the t-value is approximately 2.262.
The confidence interval formula is:
CI = x̄ ± t*(s/√n)
Plugging in the values, we get:
CI = 3.8 ± 2.262*(1.623/√10)
CI = 3.8 ± 1.316
CI = (2.484, 5.116)
Therefore, we can be 95% confident that the true average number of hours per day adults spend in watching television is between 2.484 and 5.116 hours.
To construct a 95% confidence interval to estimate the average number of hours per day adults spend watching television, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / √sample size)
First, we need to find the sample mean and the standard deviation.
Step 1: Find the sample mean
The sample mean is simply the average of the given data.
Sample mean = (2 + 5 + 4 + 6 + 7 + 4 + 2 + 3 + 2 + 3) / 10 = 38 / 10 = 3.8
Step 2: Find the standard deviation
The standard deviation measures the spread of the data.
Standard deviation (σ) = √[Σ(xᵢ - μ)² / N], where Σ is the summation symbol, xᵢ is each data point, μ is the population mean (which we estimate using the sample mean), and N is the sample size.
Calculating the standard deviation:
(2-3.8)² + (5-3.8)² + (4-3.8)² + (6-3.8)² + (7-3.8)² + (4-3.8)² + (2-3.8)² + (3-3.8)² + (2-3.8)² + (3-3.8)² = 42.72
σ = √(42.72 / 9) = 2.073
Step 3: Find the critical value
Since we are constructing a 95% confidence interval, we need to find the critical value for a 2-tailed test with a confidence level of 95%. This critical value corresponds to a z-score.
Using a standard normal distribution table or calculator, the critical value for a 95% confidence level is approximately ±1.96.
Step 4: Calculate the confidence interval
Using the formula mentioned earlier, we can now calculate the confidence interval:
Confidence Interval = 3.8 ± (1.96 * 2.073 / √10) = 3.8 ± 1.440
Therefore, the 95% confidence interval to estimate the average number of hours per day adults spend watching television is (2.36, 5.24).