One of your friends knows that you are taking Advanced Placement Statistics and asks for your help wither Chemistry lab report. She has come up with five measurements of the melting point of a compound: 122.4, 121.8, 122.0, 123.0, and 122.3 degrees Celsius.
a. The lab manual asks for a 95% confidence interval estimate for the melting point. Show her how to find this estimate.
b. Explain to your friend in simple language what 95% confidence interval means.
c. Would a 90% confidence interval be narrower or wider than the 95% interval. Explain your answer to your friend.
d. The lab manual asks whether the data show sufficient evidence to call into question the established melting point of 122 degrees Celsius. State the null and alternative hypotheses and find the P-value.
e. Explain the meaning of the specific P-value to your friend.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
However, I will start you out.
95% = mean ± Z(SEm)
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.475) and its Z score.
a. To find a 95% confidence interval estimate for the melting point, you can follow these steps:
1. Calculate the mean of the measurements by adding up all the measurements and dividing by the total number of measurements (in this case, 5).
- Mean = (122.4 + 121.8 + 122.0 + 123.0 + 122.3) / 5 = 611.5 / 5 = 122.3 degrees Celsius.
2. Calculate the standard deviation of the measurements.
- Subtract the mean from each measurement and square the result.
(122.4 - 122.3)^2 = 0.01
(121.8 - 122.3)^2 = 0.25
(122.0 - 122.3)^2 = 0.09
(123.0 - 122.3)^2 = 0.49
(122.3 - 122.3)^2 = 0
- Add up all the squared differences and divide by the total number of measurements minus 1 (in this case, 4).
(0.01 + 0.25 + 0.09 + 0.49 + 0) / 4 = 0.84 / 4 = 0.21
- Take the square root of the result.
√(0.21) ≈ 0.458
3. Calculate the margin of error using the following formula:
Margin of Error = Critical value x Standard deviation / √(n)
- The critical value for a 95% confidence interval is approximately 1.96.
- Substituting the values into the formula:
Margin of Error = 1.96 x 0.458 / √(5) ≈ 0.47
4. Calculate the lower and upper bounds of the confidence interval:
Lower bound = Mean - Margin of Error
= 122.3 - 0.47 ≈ 121.83 degrees Celsius
Upper bound = Mean + Margin of Error
= 122.3 + 0.47 ≈ 122.77 degrees Celsius
Therefore, the 95% confidence interval estimate for the melting point is approximately 121.83 to 122.77 degrees Celsius.
b. A 95% confidence interval means that if we were to repeat the experiment multiple times and construct 95% confidence intervals, approximately 95% of those intervals would contain the true population parameter, which in this case is the true melting point. It provides a range of values within which we can be reasonably confident that the true value falls.
c. A 90% confidence interval would be narrower than the 95% confidence interval. This is because the level of confidence is lower, so we allow for a smaller margin of error. The critical value for a 90% confidence interval is smaller than the critical value for a 95% confidence interval, resulting in a narrower range of values.
d. To answer whether the data show sufficient evidence to call into question the established melting point of 122 degrees Celsius, we need to conduct a hypothesis test. The null hypothesis (H0) assumes that there is no significant difference between the observed data and the established melting point. The alternative hypothesis (Ha) assumes that there is a significant difference.
Null Hypothesis (H0): The true melting point is equal to 122 degrees Celsius.
Alternative Hypothesis (Ha): The true melting point is not equal to 122 degrees Celsius.
To find the P-value, we would compare the test statistic (calculated from the sample data) to a critical value or use statistical software. The specific steps depend on the test being conducted.
e. The P-value represents the probability of obtaining the observed sample data (or more extreme) if the null hypothesis is true. In other words, it measures the strength of evidence against the null hypothesis. A smaller P-value suggests stronger evidence against the null hypothesis, while a larger P-value suggests weaker evidence. In general, if the P-value is less than the predetermined significance level (such as 0.05), we would reject the null hypothesis and conclude that there is sufficient evidence to call into question the established melting point.