answer:To test the research hypothesis that women will like Royal Dansk cookies more than men, we can perform an independent samples t-test. Here's how you would conduct the test with the scores provided (assuming equal variances): First, calculate the means for both groups: Men: (45 + 55 + 27 + 40 + 71 + 60 + 5 + 25 + 75 + 35) / 10 = 43.8 Women: (84 + 56 + 70 + 80 + 45 + 30 + 90 + 20 + 75 + 80) / 10 = 63 Next, calculate the sample variances: Men: s² = Σ(Xi - X̄)² / (n-1) Women: s² = Σ(Yi - Ȳ)² / (n-1) Men: Σ(Xi - 43.8)² = (1.2)² + (11.2)² + (-16.8)² + (-3.8)² + (27.2)² + (16.2)² + (-38.8)² + (-18.8)² + (31.2)² + (-8.8)² = 1.44 + 125.44 + 282.24 + 14.44 + 739.84 + 262.44 + 1505.44 + 353.44 + 973.44 + 77.44 s²_men = (4035.6) / (10 - 1) s²_men ≈ 448.4 Women: Σ(Yi - 63)² = (21)² + (-7)² + (7)² + (17)² + (-18)² + (-33)² + (27)² + (-43)² + (12)² + (17)² = 441 + 49 + 49 + 289 + 324 + 1089 + 729 + 1849 + 144 + 289 s²_women = (5251) / (10 - 1) s²_women ≈ 583.4 Then calculate the pooled variance: s²_pooled = ((n1 - 1) * s²_men + (n2 - 1) * s²_women) / (n1 + n2 - 2) s²_pooled = ((10 - 1) * 448.4 + (10 - 1) * 583.4) / (10 + 10 - 2) s²_pooled = (9 * 448.4 + 9 * 583.4) / 18 s²_pooled ≈ (8,124.4) / 18 s²_pooled ≈ 451.36 Now, compute the standard error of the difference between the two means: SE = √(s²_pooled * (1/n1 + 1/n2)) SE = √(451.36 * (1/10 + 1/10)) SE = √(451.36 * 0.2) SE ≈ √90.272 SE ≈ 9.499 Next, calculate the t-statistic: t = (X̄ - Ȳ) / SE t = (43.8 - 63) / 9.499 t = -19.2 / 9.499 t ≈ -2.021 Next, calculate the degrees of freedom for the t-distribution: df = n1 + n2 - 2 df = 10 + 10 - 2 df = 18 Finally, we would compare the calculated t-value to a critical t-value from a t-distribution table at the desired significance level (usually 0.05 for a two-tailed test in social sciences), but since the research hypothesis predicts a specific direction (women > men), you could use a one-tailed test. Considering the mean score for women (63) is higher than for men (43.8), and if the t-statistic is found to be greater than the critical t-value with 18 degrees of freedom, we would reject the null hypothesis in favor of the alternative hypothesis that women like Royal Dansk cookies more than men. Now, with the given data and the positive direction of the hypothesis, the calculated negative t-value implies that the mean for women is indeed higher than that for men. Depending on the critical t-value from the t-table for 18 degrees of freedom with your chosen alpha level, you may conclude to reject the null hypothesis or not. More refined statistical software or a more detailed table would give a precise p-value for this t-statistic, allowing you to make a conclusion. Note that in a real analysis, you would also want to check assumptions like normality and equal variance (with a test like Levene's test) before proceeding with a t-test. If these assumptions are violated, other statistical tests might be more appropriate.

answer:To conduct a t-test for independent samples, we should follow these steps: 1. **State the hypotheses.** - **Null Hypothesis (H0):** The mean score for men (μ1) is equal to the mean score for women (μ2). In other words, women do not like Royal Dansk cookies more than men. - **Alternative Hypothesis (H1):** The mean score for women (μ2) is greater than the mean score for men (μ1). Women like Royal Dansk cookies more than men. 2. **Calculate the mean for each group.** - Men's Mean (X̄1) = (45 + 55 + 27 + 40 + 71 + 60 + 5 + 25 + 75 + 35) / 10 = 438 / 10 = 43.8 - Women's Mean (X̄2) = (84 + 56 + 70 + 80 + 45 + 30 + 90 + 20 + 75 + 80) / 10 = 630 / 10 = 63.0 3. **Calculate the variances and standard deviation for each group.** To calculate variance, use the formula: s² = Σ(Xi - X̄)² / (n-1) For Men: s²_men = [(45-43.8)² + (55-43.8)² + (27-43.8)² + ... + (35-43.8)²] / (10 - 1) s²_men = [(1.2)² + (11.2)² + (-16.8)² + ... + (-8.8)²] / (10 - 1) s²_men = [1.44 + 125.44 + 282.24 ... + 77.44] / 9 s²_men = [sum of squared differences] / 9 Calculate this sum to find the variance. For Women: s²_women = [(84-63)² + (56-63)² + ... + (80-63)²] / (10 - 1) s²_women = [sum of squared differences] / 9 Calculate this sum to find the variance. 4. **Calculate the pooled variance.** Use the formula: s²_pooled = ((n1 - 1) * s²_men + (n2 - 1) * s²_women) / (n1 + n2 - 2) 5. **Calculate the standard error of the mean difference.** Use the formula: SE_difference = √(s²_pooled * (1/n1 + 1/n2)) 6. **Calculate the t-statistic.** Use the formula: t = (X̄1 - X̄2) / SE_difference 7. **Determine the degrees of freedom.** df = n1 + n2 - 2 8. **Determine the critical t-value from the t-distribution table.** - Since the alternative hypothesis suggests that women will like the cookies more, this is a one-tailed test. You need to check the t-distribution table for a chosen α-level (commonly 0.05) and the degrees of freedom. 9. **Make a decision.** - If the calculated t-value is more extreme than the critical t- value, you reject the null hypothesis in favor of the alternative hypothesis. Let's do the necessary calculations for our data: **Variances** For Men: Standard deviation (with decimals for precision) s²_men = [(1.2)² + (11.2)² + (-16.8)² + ... + (-8.8)²] / 9 s²_men = ≈ 1032.4 / 9 s²_men ≈ 114.7 (rounded to one decimal place) For Women: Standard deviation (with decimals for precision) s²_women = [(21)² + (-7)² + (7)² + ... + (17)²] / 9 s²_women = ≈ 2041 / 9 s²_women ≈ 226.8 (rounded to one decimal place) **Pooled Variance** s²_pooled = ((10 - 1) * 114.7 + (10 - 1) * 226.8) / (10 + 10 - 2) s²_pooled = (1032.3 + 2041.2) / 18 s²_pooled = 3073.5 / 18 s²_pooled ≈ 170.8 (rounded to one decimal place) **Standard Error** SE_difference = √(s²_pooled * (1/10 + 1/10)) SE_difference = √(170.8 * 0.2) SE_difference ≈ √34.16 SE_difference ≈ 5.8 (rounded to one decimal place) **t-Statistic** t = (X̄1 - X̄2) / SE_difference t = (43.8 - 63) / 5.8 t = -19.2 / 5.8 t ≈ -3.31 (rounded to two decimal places) **Degrees of Freedom** df = 10 + 10 - 2 = 18 To reject the null hypothesis using a one-tailed test at an α-level of 0.05, we would expect the t-value to be greater than t-critical. Let's check a t-table for df = 18: the critical t-value for a one-tailed test at α = 0.05 is approximately 1.734. Our calculated t-value is -3.31, which is more extreme than -1.734 (remember that t-distributions are symmetrical). Therefore, we reject the null hypothesis, supporting the research hypothesis that women like Royal Dansk cookies more than men. Please note: The variance and standard deviation were roughly calculated. You'd want to use either a calculator or statistical software for precise calculations including the exact decimals.

answer:In hypothesis testing with t-tests, the variance of the sampling distribution of the mean, more formally known as the "standard error of the mean" (SEM), is used to determine how much the mean scores of the samples we're comparing might vary from the actual population means. To calculate the standard error of the mean for both male and female groups, we'll need to first calculate the variance for each group based on their scores. Let's calculate the variances and then the standard error of the mean for each group. Here are the steps for calculating the variance (which has already been done in previous instructions but for clarity, I'll repeat the process). First, recalculate the means: - Men's Mean (M̄) = 43.8 - Women's Mean (W̄) = 63.0 Then calculate the variance for each group. The formula for variance (s²) is: [ s^2 = frac{sum (X_i - bar{X})^2}{n - 1} ] For men: [ s^2_{men} = frac{(45-43.8)^2 + (55-43.8)^2 + dots + (35-43.8)^2}{10 - 1} ] For women: [ s^2_{women} = frac{(84-63)^2 + (56-63)^2 + dots + (80-63)^2}{10 - 1} ] Where: ( X_i ) = each individual score, ( bar{X} ) = the mean of the scores, ( n ) = the number of scores. Let's use approximate values that we calculated earlier: - ( s^2_{men} approx 114.7 ) - ( s^2_{women} approx 226.8 ) To calculate the standard error of the mean, use the formula: [ SE = sqrt{frac{s^2}{n}} ] Standard Error for men: [ SE_{men} = sqrt{frac{s^2_{men}}{n_{men}}} ] [ SE_{men} = sqrt{frac{114.7}{10}} ] [ SE_{men} = sqrt{11.47} ] [ SE_{men} approx 3.39 ] Standard Error for women: [ SE_{women} = sqrt{frac{s^2_{women}}{n_{women}}} ] [ SE_{women} = sqrt{frac{226.8}{10}} ] [ SE_{women} = sqrt{22.68} ] [ SE_{women} approx 4.76 ] Remember that these are the standard errors of the means for each individual group, not the standard error for the difference between the two means which we calculated in the previous steps for the t-statistic. The variance of the distribution of means (the square of the standard error) would therefore be: - ( Var_{men} = SE_{men}^2 approx 3.39^2 approx 11.47 ) - ( Var_{women} = SE_{women}^2 approx 4.76^2 approx 22.68 ) These variances represent the variability of the sample means if we were to take many samples of the same size from the population of men and women, respectively, and calculate their means.

answer:The standard deviation of the distribution of differences between two sample means, also referred to as the standard error of the difference between means, quantifies how much we would expect the difference between the sample means to vary due to random sampling. To calculate the standard error of the difference between the two means (which is what we need for the t-test), we would use the following formula: [ SE_{diff} = sqrt{frac{s^2_1}{n_1} + frac{s^2_2}{n_2}} ] where: - ( s^2_1 ) is the variance for the first sample (men in this case), - ( n_1 ) is the number of observations in the first sample, - ( s^2_2 ) is the variance for the second sample (women in this case), and - ( n_2 ) is the number of observations in the second sample. Given the approximate variances we calculated earlier: - ( s^2_{men} approx 114.7 ) - ( s^2_{women} approx 226.8 ) - and the sample sizes are both 10 (since there are 10 scores in each group) Let's calculate the standard error of the difference between the sample means: [ SE_{diff} = sqrt{frac{114.7}{10} + frac{226.8}{10}} ] [ SE_{diff} = sqrt{11.47 + 22.68} ] [ SE_{diff} = sqrt{34.15} ] [ SE_{diff} approx 5.84 ] Therefore, the standard deviation of the distribution of differences between means (standard error of the difference between means) is approximately 5.84. Please remember the actual values should be calculated using precise figures rather than the approximations I used here for illustration. The actual variances (s²_men and s²_women) should be calculated using the full precision of the data in order to achieve the most accurate results. If you have access to statistical software, it can compute these values to full precision and also provide the exact t-statistic and corresponding p-value, along with any assumptions that need to be tested, like the equality of variances, which would be checked using something like Levene's test.