Question

The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The average age in the sample was 22.5 years with a standard deviation of 5 years. a) Construct a 95% confidence interval for the true average age of the consumers. b) Construct an 80% confidence interval for the true average age of the consumers. c) Discuss why the 95% and 80% confidence intervals are different.

Answer 1

a) The 95% confidence interval is (19.543, 25.457). b) The 80% confidence interval is (20.708, 24.292). c) The 95% and 80% confidence intervals are different due to the different critical values used.

Step-by-step explanation:

a) Constructing a 95% confidence interval:

To construct a 95% confidence interval for the true average age of the consumers, we can use the formula:

Lower limit = sample mean - (critical value * standard error)

Upper limit = sample mean + (critical value * standard error)

Where the critical value is based on the desired level of confidence and the sample size. For a 95% confidence level with 16 consumers, the critical value is approximately 2.131. The standard error can be calculated by dividing the standard deviation by the square root of the sample size.

Using the given values, we have:

Lower limit = 22.5 - (2.131 * (5 / sqrt(16)))

Upper limit = 22.5 + (2.131 * (5 / sqrt(16)))

Calculating these values, we get:

Lower limit = 19.543

Upper limit = 25.457

Therefore, the 95% confidence interval for the true average age of the consumers is (19.543, 25.457).

b) Constructing an 80% confidence interval:

To construct an 80% confidence interval, we follow the same steps as in part (a), but use a different critical value. For an 80% confidence level with 16 consumers, the critical value is approximately 1.341. Using this critical value, we calculate the lower and upper limits as:

Lower limit = 22.5 - (1.341 * (5 / sqrt(16)))

Upper limit = 22.5 + (1.341 * (5 / sqrt(16)))

Calculating these values, we get:

Lower limit = 20.708

Upper limit = 24.292

The 80% confidence interval for the true average age of the consumers is (20.708, 24.292).

c) Difference between the 95% and 80% confidence intervals:

The 95% and 80% confidence intervals are different because they use different critical values. A higher confidence level requires a larger critical value, resulting in a wider interval. In other words, the 95% confidence interval provides a higher level of certainty in capturing the true average age compared to the 80% confidence interval.

Answer 2

Answer:

Step-by-step explanation:

Assuming a normal distribution for the age of the consumers. We want to determine a 95% confidence interval for the true average age of the consumers.

Number of sample, n = 16

Mean, u = 22.5 years

Standard deviation, s = 5 years

For a confidence level of 95%, the corresponding z value is 1.96. This is determined from the normal distribution table.

We will apply the formula

Confidence interval

= mean ± z ×standard deviation/√n

It becomes

22.5 ± 1.96 × 5/√16

= 22.5 +/- 1.96 × 1.25

= 22.5 +/- 2.45

The lower end of the confidence interval is 22.5 - 2.45 =20.05

The upper end of the confidence interval is 22.5 + 2.45 =24.95

For 80% confidence interval,

the corresponding z value is 1.28. This is determined from the normal distribution table.

We will apply the formula

Confidence interval

= mean ± z ×standard deviation/√n

It becomes

22.5 ± 1.28 × 5/√16

= 22.5 +/- 1.28× 1.25

= 22.5 +/- 1.6

The lower end of the confidence interval is 22.5 - 1.6 =20.9

The upper end of the confidence interval is 22.5 + 1.6 =24.1

95% provides a wider interval than 80%. The wider the possible values of the true mean, the more confident we are. The lesser the possible values of the true mean, the less confident we are