Z and T Statistics: Confidence Intervals and Significance Tests:
A friend who hears that you are taking a statistics course asks for help with a
specific chemistry lab report. She has made four independent measurements of the
specific gravity of a compound. The results are : 4.48, 4.6, 4.33 and 4.57. You are
willing to assume that the measurements are not biased. This means that the mean M of the
distribution of measurement is the true specific gravity.
4.48
4.6
4.33
4.57
a) Calculate a 95% confidence interval for the true specific gravity for your friend.
Use Table T
b) Explain to your friend what this means
Use Table T
c) What must be true about your friend's measurements for your results in part (a) to be
correct?
Use Table T
d) You notice that the lab manual says that repeated measurements will vary according to a normal
distribution with standard deviation 0=0.11. Redo the confidence interval of part (a) using this additional
information. Explain why we expect the new interval to be shorter.
Use Table Z*
e) What critical value from the table would you use for an 80% confidence interval? Without
calculating that interval would you expect it to be wider or narrower that the 95% confidence interval?
Use Table Z*
f) The lab manual also asks whether the data show convincingly that the true specific gravity is less
than 4.5. State the null hypothesis used to answer this question. Then calculate the test statistic and
find its P-value. Use the lab manual's value o=0.11 and calculate the p-value in detail.
Use Table Z*
g) Explain to your friend what your p-value means.
Use Table Z*
a) To calculate a 95% confidence interval for the true specific gravity, we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)
First, we need to calculate the sample mean by taking the average of the four measurements: (4.48 + 4.6 + 4.33 + 4.57) / 4 = 4.495.
Next, we need to calculate the standard error, which is the standard deviation divided by the square root of the sample size. Since the sample size is 4 and the standard deviation is not given, we'll assume it's the population standard deviation. Therefore, the standard error is 0.11 / sqrt(4) = 0.055.
Now, we need to find the critical value. Since the sample size is small (less than 30), we'll use the t-distribution with 4 - 1 degrees of freedom. Looking up the critical value in the t-table at a 95% confidence level and 3 degrees of freedom, we find the value to be approximately 3.182.
Finally, we can calculate the confidence interval using the formula:
Confidence interval = 4.495 ± (3.182 * 0.055) = (4.37, 4.62).
Therefore, the 95% confidence interval for the true specific gravity is (4.37, 4.62).
b) This means that we are 95% confident that the true specific gravity falls within the range of 4.37 to 4.62 based on the sample data.
c) For the results in part (a) to be correct, we assume that the measurements are not biased and that they are independent. In other words, each measurement was made under the same conditions, and there is no systematic error or bias present in the measurements.
d) To redo the confidence interval using the additional information provided (standard deviation = 0.11), we can use the same formula as in part (a), but now we'll use the z-distribution instead of the t-distribution.
The standard error remains the same: 0.11 / sqrt(4) = 0.055.
Using the z-table, we can find the critical value at a 95% confidence level, which is approximately 1.96.
Now, we can calculate the confidence interval using the formula:
Confidence interval = 4.495 ± (1.96 * 0.055) = (4.39, 4.60).
Since the z-distribution assumes a larger sample size, the confidence interval is expected to be shorter compared to the t-distribution used in part (a).
e) To find the critical value from the table for an 80% confidence interval, we need to find the z-score that corresponds to a cumulative probability of 0.80. Looking up the z-table, we find that the critical value is approximately 1.282.
Without calculating the interval, we expect the 80% confidence interval to be narrower than the 95% confidence interval. This is because a higher confidence level requires a wider interval to capture a larger range of possible values.
f) The null hypothesis is a statement which assumes no effect or no difference. In this case, the null hypothesis would be: "The true specific gravity is equal to 4.5."
To calculate the test statistic, we use the formula:
Test statistic = (sample mean - null hypothesis mean) / (standard deviation / sqrt(sample size))
= (4.495 - 4.5) / (0.11 / sqrt(4))
= -0.005 / (0.11 / 2)
= -0.005 / 0.055
= -0.0909
Next, we need to find the p-value, which represents the probability of obtaining a test statistic equal to or more extreme than the one observed, assuming the null hypothesis is true.
Using the z-table, we find that the p-value for a test statistic of -0.0909 is approximately 0.4638.
Therefore, the p-value for the test statistic is 0.4638.
g) The p-value represents the probability of obtaining a test statistic as extreme (or more extreme) as the one observed, assuming the null hypothesis is true. In this case, a p-value of 0.4638 means that, if the true specific gravity is actually equal to 4.5, there is a 46.38% chance of obtaining a sample mean as different from 4.5 or more extreme as the one observed. Since the p-value is greater than the commonly used significance level of 0.05, we would not have enough evidence to reject the null hypothesis and conclude that the true specific gravity is less than 4.5.