5. The following random sample was selected : 4, 6, 3, 5, 9, 3. Find the 95% confidence interval for the mean of the population.
6. In a sample of 35 high school seniors, 14 of them are attending college in the fall. Find the 95% confidence interval for the true proportion of high school seniors that will attend college in the fall from the population.
7. In a sample of 200 people, 76 people would rather work out at home than in a gym. Find the 99% confidence interval for the true proportion of people who would rather work out at home than in a gym for the entire population.
8. A study found that out of 300 people 60% of them prefer to eat hamburgers rather than hot dogs. Fin the 95% confidence interval for the true proportion of people who prefer to eat hamburgers rather than hot dogs in the entire population.
5. 31.5 is the mean
5. To find the 95% confidence interval for the mean, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
First, let's calculate the sample mean. Add up all the numbers in the sample: 4 + 6 + 3 + 5 + 9 + 3 = 30. Then, divide by the number of observations in the sample, which is 6. So, the sample mean is 30/6 = 5.
Next, we need to calculate the standard deviation. To do this, we first need to find the variance. Subtract the sample mean from each observation, square the difference, and sum up all the squared differences. Then, divide by the sample size minus 1 (this is called the degrees of freedom). The calculations would be as follows:
(4 - 5)^2 + (6 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (9 - 5)^2 + (3 - 5)^2 = 8 + 1 + 4 + 0 + 16 + 4 = 33
Divide 33 by 6 - 1 = 5. So, the variance is 33/5 = 6.6.
To get the standard deviation, take the square root of the variance. The square root of 6.6 is approximately 2.57 (rounded to two decimal places).
Now, we need to determine the critical value for a 95% confidence level. We can use a standard normal distribution table or a statistical calculator for this. For a 95% confidence level, the critical value is approximately 1.96.
Finally, substitute the values into the confidence interval formula:
Confidence Interval = 5 ± (1.96 * 2.57 / √6)
Calculating this expression, we get:
Confidence Interval ≈ 5 ± (5.04 / 2.45)
Simplifying further, the 95% confidence interval for the mean of the population is approximately 2.06 to 7.94.
6. To find the 95% confidence interval for the true proportion of high school seniors attending college in the fall, we can use the formula:
Confidence Interval = sample proportion ± (critical value * √(sample proportion * (1 - sample proportion) / sample size))
First, let's calculate the sample proportion. Divide the number of high school seniors attending college in the fall (14) by the total number of seniors in the sample (35). So, the sample proportion is 14/35 ≈ 0.4 (rounded to two decimal places).
Next, we need to calculate the critical value for a 95% confidence level. Similar to the previous example, it is approximately 1.96.
Now, substitute the values into the confidence interval formula:
Confidence Interval = 0.4 ± (1.96 * √(0.4 * (1 - 0.4) / 35))
Calculating this expression, we get:
Confidence Interval ≈ 0.4 ± (0.16 / 35)
Simplifying further, the 95% confidence interval for the true proportion of high school seniors attending college in the fall from the population is approximately 0.348 to 0.452.
7. To find the 99% confidence interval for the true proportion of people who would rather work out at home than in a gym, we can use the same formula as before:
Confidence Interval = sample proportion ± (critical value * √(sample proportion * (1 - sample proportion) / sample size))
First, let's calculate the sample proportion. Divide the number of people who would rather work out at home (76) by the total number of people in the sample (200). So, the sample proportion is 76/200 = 0.38.
Next, calculate the critical value for a 99% confidence level. For a 99% confidence level, the critical value is approximately 2.58.
Now, substitute the values into the confidence interval formula:
Confidence Interval = 0.38 ± (2.58 * √(0.38 * (1 - 0.38) / 200))
Calculating this expression, we get:
Confidence Interval ≈ 0.38 ± (0.0904 / 200)
Simplifying further, the 99% confidence interval for the true proportion of people who would rather work out at home than in a gym in the entire population is approximately 0.333 to 0.427.
8. To find the 95% confidence interval for the true proportion of people who prefer to eat hamburgers rather than hot dogs, we can again use the same formula:
Confidence Interval = sample proportion ± (critical value * √(sample proportion * (1 - sample proportion) / sample size))
In this case, the sample proportion is given as 60%, which can be expressed as 0.6.
Next, calculate the critical value for a 95% confidence level, which is approximately 1.96.
Now, substitute the values into the confidence interval formula:
Confidence Interval = 0.6 ± (1.96 * √(0.6 * (1 - 0.6) / 300))
Calculating this expression, we get:
Confidence Interval ≈ 0.6 ± (0.0456 / 300)
Simplifying further, the 95% confidence interval for the true proportion of people who prefer to eat hamburgers rather than hot dogs in the entire population is approximately 0.553 to 0.647.