Mastering Normal Distributions, Sampling Distributions, and Estimation- Sample Size Calculation

Mastering Normal Distributions, Sampling Distributions, and Estimation- Sample Size Calculation

Question 1

The calculation for the sample size, given a margin of error (E) of $500, is as follows:

The confidence level is 90%, which implies a significance level of 0.10. Therefore, the critical z-value for a two-tailed test at this significance level is 1.65, as verified using a z-distribution table.

With a standard deviation ơ = $3100, the formula for determining the sample size (n) is as follows:

Substituting the values, we get:

= 105

Question 2

Statements regarding the margin of error and sample size are broken as follows:

1. Incorrect. This is because a smaller margin of error typically leads to increased confidence.
2. Correct, because the margin of error is inversely proportional to this proportional to 1 over the size, (Lakens, 2022).
3. Incorrect, as halving the margin of error would require quadrupling the sample size because of the inverse square relationship.

Question 3

The alternate hypothesis that would indicate that the mean of the x2 population is smaller than that of the x1 population is:

1. c) (H1: µ1 > µ2)

This hypothesis states that the mean of the first population µ1 is greater than the mean of the second population µ2, which is another way of saying that the mean of the second population is smaller than the mean of the first.

Question 4

The 90% confidence interval for a population mean is calculated as:

Given a sample mean () of 15, a sample size (n) of 50, z = 1.645 (for 90% confidence), and a standard deviation  of 3.4, the confidence interval would be calculated as follows:

= [14.21,15.78]

These results show a range of [14.21, 15.79],

This interval means that we are 90% confident that the true average amount of money that all people spend on lottery tickets each week falls between $14.21 and $15.79.

Question 5

Essential elements of hypothesis testing include:

1. Null and alternative hypotheses
   • The null hypothesis H₀ is a statement of no effect or difference that acts as the default assumption to be tested against the alternative hypothesis H₁ or Ha, which is the opposite claim or what we seek evidence for. The null hypothesis remains true unless evidence shows otherwise (Leppink et al., 2017).
2. Test Statistic
   • This is a standardized value computed from sample data during a hypothesis test. It indicates how far the sample statistic departs from the null hypothesis and is either used for finding the p-value or comparing it with the critical value.

• P-Value and Critical Value
  • The p-value represents the probability that if the null hypothesis were true, a test statistic as extreme as or more extreme than what was observed would be obtained; small values lead to rejection of null. The critical value is the threshold over which the test statistic must cross in order to reject null; it depends on the significance level, which is the probability of Type I error (rejecting true null) (Hupe, 2015).

1. Conclusion
   • Conclusions are drawn by comparing p-values with significance levels and test statistics with critical values. If there’s sufficient evidence against the null hypothesis, it is rejected in favor of the alternative. Otherwise, we fail to reject the null hypothesis. However, it is essential to note that failing to reject does not prove anything about the validity of our proposed explanations – it only suggests that there is no sufficient evidence from data gathered thus far.

Question 6

The researcher can use a one-tailed t-test to determine if the pulse rates of smokers are significantly higher than those of non-smokers. Here are the steps:

1. Null Hypothesis (H₀): The mean pulse rate of smokers is equal to the mean pulse rate of non-smokers. (µ1 = µ2)
2. Alternative Hypothesis (H₁): The mean pulse rate of smokers is greater than the mean pulse rate of non-smokers. (µ1 > µ2)

We have the following:

Sample size (n1 = n2) = 100

Mean of smokers (µ1) = 90

Mean of non-smokers (µ2) = 88

The standard deviation of smokers (σ1) = 5

The standard deviation of non-smokers (σ2) = 6

Significance level (α) = 0.05

The test statistic for a t-test is calculated as:

Substituting the given values into the formula, we have:

= 2.56082

This calculated t-value is compared with the critical t-value from the t-distribution table with a degree of freedom = n1 + n2 – 2 = 198 at α = 0.05.

The critical t value = 1.984 (from the t-distribution table)

t = 2.56082 > t_crit = 1.984

Since the calculated t-value is greater than the critical t-value, then we reject the null hypothesis is rejected, and conclude that smokers have a higher pulse rate than non-smokers at the 0.05 significance level.

References

Hupe, J.-M. (2015). Statistical inferences under the Null hypothesis: common mistakes and pitfalls in neuroimaging studies. Frontiers in Neuroscience, 9. https://doi.org/10.3389/fnins.2015.00018

Lakens, D. (2022). Sample size justification. Collabra: Psychology, 8(1). https://doi.org/10.1525/collabra.33267

Leppink, J., O’Sullivan, P., & Winston, K. (2017). Evidence against vs. in favor of a null hypothesis. Perspectives on Medical Education, 6(2), 115–118. https://doi.org/10.1007/s40037-017-0332-6

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Week 4 Assignment:
The following assignment will allow you to master the material on Normal Distributions, Sampling Distributions, and Estimation.

1. How large of a sample is required if we want to estimate the mean price of a five-year-old Corvette within +/- $500 with 90% confidence if the standard deviation is known to be $3100. (10 points)

2. Several factors are involved when creating confidence intervals, such as sample size, the level of confidence, and margin of error. Which of the following statements is true? (10 points)

a. For a given sample size, reducing the margin of error will mean lower confidence.
b. For a specified confidence interval. Larger samples provide smaller margins of error.
c. For a given confidence level, halving the margin of error requires a sample twice as large.

3. When conducting a test for the difference of means for two independent populations x1 and x2, what alternate hypothesis would indicate that the mean of the x2 population is smaller than that of the x1 population? (10 points)

a) H1: µ1 ‹ µ2
b) H1: µ1 ≠ µ2
c) H1: µ1 › µ2
d) H1: µ1 = µ2

4. A researcher wants to estimate the average amount of money a person spends on buying lottery tickets each week. A sample of 50 people who buy lottery tickets was found with a mean of $15 and a standard deviation of 3.4. Find a 90% confidence interval of the population mean. Explain in laymen terms what this interval represents. (10 points)

5. Discuss the basic components of a hypothesis test. (10 points)

6. Explain how to conclude when you would reject the null hypothesis based on a p-value. Give a numeric example. (10 points)

7. A medical rehabilitation foundation reports that the average cost of rehabilitation for stroke victims is $23,672. A researcher wishes to find out the average costs of rehabilitation by selecting a random sample of 35 stroke victims and finds their average cost to be $24,226. The standard deviation of the population is $3251. At an alpha level of 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $23,672? (20 points)

8. A medical researcher wishes to see whether the pulse rate of smokers is higher than the pulse rates of non-smokers. Samples of 100 smokers and 100 nonsmokers are selected. The results are shown below. Can the researcher conclude at an alpha = 0.05, that smokers have higher pulse rates than nonsmokers? (20 points)