A large furniture store has begun a new ad campaign on local television. Before the campaign, the long term average daily sales were $24,819. A random sample of 40 days during the new ad campaign gave a sample mean daily sale of x(bar)=$25,910 with sample standard deviation s=$1917. Does this indicate that the population mean daily sales is now more than $24,819? Use a 1% level of significance.
Use a one-sample z-test.
Ho: µ = 24819 ---> null hypothesis
Ha: µ > 24819 -->alternate hypothesis
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
With your data:
z = (25910 - 24819)/(1917/√40) = ?
Finish the calculation.
Check a z-table at .01 level of significance for a one-tailed test.
If the z-test statistic exceeds the critical value from the z-table, reject the null and conclude µ > 24819. If the z-test statistic does not exceed the critical value from the z-table, do not reject the null.
I hope this will help get you started.
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To determine whether the population mean daily sales is now more than $24,819, we can use a hypothesis test. The null hypothesis (H0) assumes that the population mean is equal to $24,819, while the alternative hypothesis (H1) assumes that the population mean is greater than $24,819.
Let's define our hypotheses:
H0: μ = $24,819 (population mean is equal to $24,819)
H1: μ > $24,819 (population mean is greater than $24,819)
We'll use a one-sample t-test to compare the sample mean with the hypothesized population mean. The formula to calculate the test statistic (t-value) is:
t = (x̄ - μ) / (s / √n)
where:
x̄ is the sample mean ($25,910),
μ is the hypothesized population mean ($24,819),
s is the sample standard deviation ($1917), and
n is the sample size (40).
Calculating the t-value:
t = ($25,910 - $24,819) / ($1917 / √40)
t = $1091 / (301.918 / √40)
t = $1091 / (301.918 / 6.324)
t = $1091 / 47.721
t ≈ 22.87 (rounded to two decimal places)
Next, we need to find the critical value (t-critical) using a significance level of 1% and degrees of freedom (df) equal to n - 1. Since n = 40, df = 39.
We can look up the critical value from the t-distribution table or use statistical software. In this case, the critical value is approximately 2.707.
Finally, we compare the t-value to the critical value to make a decision.
If the t-value is greater than the critical value, we reject the null hypothesis (H0) and conclude that there is evidence to support the alternative hypothesis (H1). If the t-value is less than or equal to the critical value, we fail to reject the null hypothesis and do not have sufficient evidence to support the alternative hypothesis.
In this case, the t-value (22.87) is greater than the critical value (2.707), which means we reject the null hypothesis. Therefore, we can conclude that there is evidence to suggest that the population mean daily sales is now more than $24,819 at a 1% level of significance.