Math Problem Statement
A marketing researcher wants to estimate the mean amount spent per year ($) on a web site by membership member shoppers. Suppose a random sample of 100 membership member shoppers who recently made a purchase on the web site yielded a mean amount spent of $57 and a standard deviation of $52. Is there evidence that the population mean amount spent per year on the web site by membership member shoppers is different from $51? (Use a 0.10 level of significance.) State the null and alternative hypotheses. Upper H 0: mu ▼ less than or equals greater than less than not equals equals greater than or equals enter your response here Upper H 1: mu ▼ less than equals greater than or equals less than or equals not equals greater than enter your response here (Type integers or decimals. Do not round. Do not include the $ symbol in your answer.) Part 2 Identify the critical value(s). The critical value(s) is/are enter your response here. (Type an integer or a decimal. Round to two decimal places as needed. Use a comma to separate answers as needed.) Part 3 Determine the test statistic. The test statistic, t Subscript STAT, is enter your response here. (Type an integer or a decimal. Round to two decimal places as needed.) Part 4 State the conclusion. ▼ Do not reject Reject Upper H 0. There is ▼ sufficient insufficient evidence that the population mean spent by membership member customers is different from $51. Part 5 b. Determine the p-value and interpret its meaning. The p-value is enter your response here. (Type an integer or a decimal. Round to three decimal places as needed.) Part 6 Interpret the meaning of the p-value. Select the correct answer below. A. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than $6 away from $51 if the null hypothesis is true. B. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than $6 above $51 if the null hypothesis is false. C. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than $6 below $51 if the null hypothesis is false. D. The p-value is the probability of not rejecting the null hypothesis when it is false.
Solution
Part 1: State the Null and Alternative Hypotheses
We are testing whether the population mean amount spent by membership member shoppers is different from $51, so this suggests a two-tailed test.
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Null hypothesis (H₀): The population mean is equal to $51.
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Alternative hypothesis (H₁): The population mean is not equal to $51.
Part 2: Identify the Critical Value(s)
Given that the level of significance () is 0.10, and it is a two-tailed test, we will divide the significance level equally between the two tails: for each tail.
The critical value for a two-tailed test with can be found using a t-distribution table (since the sample size is 100, degrees of freedom ).
Using the t-distribution for a two-tailed test at :
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Critical value for a two-tailed test at and 99 degrees of freedom is approximately:
Part 3: Determine the Test Statistic
The test statistic for a t-test is calculated using the following formula:
Where:
- (sample mean),
- (hypothesized population mean),
- (sample standard deviation),
- (sample size).
Now, substituting the values:
So, the test statistic is approximately:
Part 4: State the Conclusion
- The test statistic is less than the critical value .
- Since , we fail to reject the null hypothesis.
Thus, the conclusion is:
There is insufficient evidence that the population mean spent by membership member customers is different from $51.
Part 5: Determine the P-value
The p-value corresponds to the area under the t-distribution curve beyond the test statistic in both tails (since it's a two-tailed test).
From a t-distribution table or using statistical software:
Part 6: Interpret the Meaning of the P-value
The p-value is the probability of observing a sample mean as extreme as $6 away from $51 (either above or below) if the null hypothesis is true. The correct interpretation is:
Option A:
The p-value is the probability of obtaining a sample mean that is equal to or more extreme than $6 away from $51 if the null hypothesis is true.
Do you want more details or have any questions?
Here are 5 related questions to consider:
- What is the difference between one-tailed and two-tailed tests?
- How do you determine the degrees of freedom for a t-test?
- What does the critical value represent in hypothesis testing?
- How does sample size affect the outcome of a t-test?
- What is the significance level, and how do you choose it?
Tip: Always check whether you're performing a one-tailed or two-tailed test, as it directly affects the critical values and p-value interpretation.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Two-tailed t-test
Statistical Significance
P-value
Formulas
t_stat = (x̄ - μ₀) / (s / √n)
Critical value for t-distribution
Theorems
Central Limit Theorem
Student's t-distribution
Suitable Grade Level
Undergraduate (Statistics or Business Analytics)
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