1)According to past record the average daily sales for a sporting goods store was $3456.00. After a new advertising campaign, the manager wants to see if the average daily sales have changed. A random sample of 20 days gives a sample mean of 3655 with a standard deviation of $458. Daily sales are assumed to be distributed normal.
a)Explain the underlying population under consideration
b)Write down the given information using correct symbols
3456= ------------------; 3655= -----------------------; 458= -----------------
c)Test to see if there is enough evidence to show that the average daily sale after the advertising is changed at 5% level of significance.
d)Find the p-value and interpret it; P-value = --------; Interpretation:
a) The underlying population under consideration is the daily sales for a sporting goods store. It represents the total sales made by the store on a daily basis.
b) Given information using correct symbols:
- Population mean (before advertising campaign): μ = $3456
- Sample mean (after advertising campaign): x̄ = $3655
- Sample standard deviation: σ = $458
c) To test if there is enough evidence to show that the average daily sale has changed after the advertising campaign, we can use a hypothesis test. The null hypothesis (H0) assumes that the average daily sales have not changed, while the alternative hypothesis (Ha) assumes that the average daily sales have changed.
The hypotheses for this test are:
H0: μ = $3456 (no change in average daily sales)
Ha: μ ≠ $3456 (change in average daily sales)
To test these hypotheses, we can use a t-test since the population standard deviation (σ) is not known. By using the sample data, we can calculate the test statistic and compare it to the critical value at a significance level of 5%.
d) The p-value is a measure of the strength of evidence against the null hypothesis. In this case, the p-value represents the probability of observing a sample mean of $3655 (or more extreme) if the true population mean is $3456 (assuming the null hypothesis is true).
To find the p-value, we can use the t-distribution with n-1 degrees of freedom, where n is the sample size. By calculating the t-score using the formula:
t = (x̄ - μ) / (σ / √(n)), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Once the t-score is calculated, the p-value can be determined by finding the probability of obtaining a t-score as extreme or more extreme than the calculated t-score from the t-distribution.
Interpretation: If the p-value is less than the significance level (in this case, 0.05 or 5%), it means that the observed sample mean is significantly different from the population mean of $3456. This indicates enough evidence to reject the null hypothesis and conclude that the average daily sales have changed after the advertising campaign. On the other hand, if the p-value is greater than 0.05, it means that we do not have enough evidence to reject the null hypothesis and conclude that there has been a change in the average daily sales.