We want a confidence interval for the mean IQ of all the 7th grade girls in that school district. We do not know the population standard deviation so we use the confidence interval that uses the t. In order to use the t-interval we need to assume that the variable IQ has a normal distribution.
1. An important condition is that these 31 girls are an ______ of all seventh-grade girls in the school district.
2. Produce a histogram for IQ and tell me how you feel about using the t confidence interval or the t test.
a. I am terribly worried the distribution is very skewed and the sample size is very small, I should not use the t.
b. It is OK to use the t, the distribution is fairly symmetric, except for an outlier in the lower tail, but the sample size is not too small thus I trust the robustness of the t-procedure.
c. I should be using the formula for proportions instead.
3. Use STAT > Basic Statistics > 1 sample t to get a 95% confidence interval for the population mean. We get the following interval __________. Which of these interpretations is closest to the truth?
a. 95% of all the girls in 7th grade in the school district have an IQ between 100.6 and 111.07.
b. 95% of the 31 girls in the sample have an IQ between 100.60 and 111.07.
c. We are 95% confident that the mean IQ of all the girls in 7th grade in the school district is between 100.60 and 111.07.
d. There is a 0.95 probability that all the girls in the 7th grade school district have an IQ between 100.60 and 111.07.
e. We are 95% confident that the mean IQ for the 31 girls in the sample is between 100.60 and 111.07.
4. Imagine we want to test the hypothesis that H0: µ = 100 vs. Ha: µ ≠ 100 using α = 0.05. Use the confidence interval to make a decision about the null hypothesis.
a. We can’t use the confidence interval to a make a decision.
b. We would reject the null hypothesis because the 100 is not in the confidence interval.
c. We would NOT reject the null hypothesis.
d. I don’t have a p-value so I can’t make a decision.
5. How would you describe the situation in the previous question?
a. It is a clear case of statistical significance and practical significance, the population mean is very far from 100.
b. This is likely to be a case of statistical significance, because we reject the null hypothesis, but not of practical significance because the lower end of the interval 100.6 is quite close to 100.
c. This is a clear case of practical significance but not statistical significance.
1. statistic
2. B.
3.
3. 111.07
and i think E
1. An important condition is that these 31 girls are a representative sample of all seventh-grade girls in the school district.
To ensure that the sample is representative, it should be randomly selected from the population of all seventh-grade girls in the school district. This means that every girl in the population has an equal chance of being included in the sample. Random sampling helps to minimize any bias and increase the likelihood that the sample is an accurate representation of the population.
2. To assess the suitability of using a t confidence interval or t-test, we can produce a histogram for IQ and evaluate its shape and sample size.
a. If the histogram shows a highly skewed distribution and the sample size is small, it may not be appropriate to use the t-confidence interval or t-test. In this case, alternative methods, such as non-parametric tests, may be more appropriate.
b. If the histogram shows a fairly symmetric distribution, with the exception of some outliers, and the sample size is not too small, it is generally acceptable to use the t-confidence interval or t-test. The t-procedure is robust enough to handle moderate deviations from normality and is still reliable with larger sample sizes.
c. The formula for proportions would be used if the variable of interest were a categorical variable with two levels (e.g., yes/no, pass/fail), rather than a continuous variable like IQ.
3. To obtain a 95% confidence interval for the population mean, we can use the "1 sample t" function in statistical software.
The resulting interval will depend on the specific data, but the correct interpretation is:
c. We are 95% confident that the mean IQ of all the girls in 7th grade in the school district is between 100.60 and 111.07.
This means that if we repeated the sampling process many times and constructed a 95% confidence interval for each sample, approximately 95% of those intervals would contain the true population mean.
4. We can use the confidence interval to make a decision about the null hypothesis.
The correct answer is:
b. We would reject the null hypothesis because 100 is not in the confidence interval.
Since the null hypothesis assumes that the population mean (µ) is equal to 100, if the hypothesized value of 100 falls outside the confidence interval, it suggests that the null hypothesis is unlikely to be true. Thus, we would reject the null hypothesis in favor of the alternative hypothesis (Ha: µ ≠ 100).
5. The situation in the previous question involves both statistical and practical significance.
The correct answer is:
b. This is likely to be a case of statistical significance because we reject the null hypothesis, but not of practical significance because the lower end of the interval (100.6) is quite close to 100.
Statistical significance refers to the likelihood of observing the obtained data if the null hypothesis were true. In this case, rejecting the null hypothesis implies that the observed data is unlikely to have occurred by chance alone. However, practical significance refers to the magnitude of the observed effect in a real-world context. Since the confidence interval includes values very close to 100, it indicates that the difference may not have practical significance in the context of IQ scores.