1. Suppose the estimate of a proportion of a normal population is to be within 0.05 using the 95% confidence level. How large a sample is required if:
a. It is known from previous studies that p = 0.3?
b. There is no prior knowledge of p?
2. Each of the following is a large-sample confidence interval for μ, the mean resonance frequency (HZ) for all tennis rackets of a certain type. (114.4, 115.6), (114.1, 115.9)
a. What is the value of the sample mean resonance frequency?
b. Both intervals were calculated from the same data. The confidence level of one of these intervals is 90% and the other is 99%. Which of these intervals has the 99% confidence level, and why?
3. Determine the t-critical value for a two-sided confidence interval in each of the following situations.
a. Confidence level = 95%, df = 15
b. Confidence level = 99%, n = 5.
c. Confidence level = 95%, n = 15
d. Significance level = 0.01, df = 37
4. A study of the ability of individuals to walk in a straight line (“Can We Really Walk Straight?”, Amer. J. of Physical Anthro., 1992:19-27) reported the accompanying data on cadence (strides per second) for a sample of 20 randomly selected individuals.
0.95 0.82 0.92 0.95 0.93 0.86 1.00 0.92 0.85 0.81 0.78 0.93 1.05 0.93 1.06 0.96 0.81 0.96 0.92 0.93
Construct a 95% confidence interval for the true mean cadence.
5. An Aspirin manufacturer fills bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a large lot is weighed, resulting in a sample average weight per tablet of 4.87 grains and a sample standard deviation of 0.35 grain. Does this information provide strong evidence for concluding that the company is not filling its bottles as advertised? Test H0: μ = 5 against HA: μ < 5 at the 0.01 level of significance
1.
a. To determine the sample size required when the population proportion is known (p=0.3), we can use the formula for sample size calculation for estimating a proportion:
n = (Z^2 * p * (1 - p)) / (E^2)
where:
- n is the required sample size
- Z is the Z-score corresponding to the desired confidence level (95% in this case)
- p is the estimated population proportion
- E is the desired margin of error (0.05 in this case)
Plugging in the values, we have:
n = (1.96^2 * 0.3 * (1 - 0.3)) / (0.05^2)
n = 345.6
Therefore, a sample size of at least 346 is required when p = 0.3.
b. When there is no prior knowledge of the population proportion (p), we assume the most conservative estimate of p, which is 0.5. Using the same formula as above, we can calculate the required sample size:
n = (1.96^2 * 0.5 * (1 - 0.5)) / (0.05^2)
n = 384.16
Therefore, a sample size of at least 385 is required when there is no prior knowledge of p.
2.
a. To find the sample mean resonance frequency, we simply take the average of the upper and lower bounds of the confidence interval:
Sample mean = (114.4 + 115.6) / 2 = 115
b. In a confidence interval calculation, a higher confidence level indicates a wider interval. Therefore, the interval (114.1, 115.9) has the 99% confidence level, while the interval (114.4, 115.6) has the 90% confidence level. This is because the wider interval allows for a higher level of confidence in capturing the true population mean.
3.
a. To determine the t-critical value for a two-sided confidence interval with a 95% confidence level and df = 15, we use the t-distribution table or a calculator. Looking up the values, the t-critical value is approximately 2.131.
b. For a 99% confidence level and n = 5, we use a t-critical value of approximately 4.604.
c. For a 95% confidence level and n = 15, the t-critical value is approximately 2.13.
d. To find the t-critical value for a significance level of 0.01 and df = 37, we use the t-distribution table or a calculator. The t-critical value is approximately 2.713.
4. To construct a 95% confidence interval for the true mean cadence, we can use the sample data provided. We can calculate the sample mean (x̄) and the standard error (SE) using the following formulas:
Sample mean (x̄) = Σx / n
Standard error (SE) = s / √n
where:
- Σx is the sum of all the individual cadence values
- n is the number of observations (in this case, 20)
- s is the sample standard deviation
After calculating x̄ and SE, we can construct the confidence interval using the formula:
Confidence interval = x̄ ± (t * SE)
where t is the t-critical value for a 95% confidence level and df = n - 1.
5. To test whether the company is not filling the bottles as advertised, we can perform a hypothesis test. The null hypothesis (H0) states that the mean weight per tablet is equal to 5 grains (μ = 5), while the alternative hypothesis (HA) states that the mean weight per tablet is less than 5 grains (μ < 5).
We can calculate the t-test statistic using the formula:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean weight per tablet, μ is the hypothesized population mean (5 grains), s is the sample standard deviation, and n is the sample size (100 tablets).
We can then compare the t-test statistic to the critical value for a one-tailed t-test with a significance level of 0.01 and degrees of freedom (df) = n - 1. If the t-test statistic is less than the critical value, we can reject the null hypothesis and conclude that there is evidence that the company is not filling its bottles as advertised.
Note: The critical value for a one-tailed test at a significance level of 0.01 with df = 99 can be found using the t-distribution table or a calculator.